messages
listlengths 1
1
| ground_truth
listlengths 15
15
| dataset
stringclasses 1
value | id
stringlengths 5
8
| difficulty
int64 4
4
| timestamps
listlengths 15
15
| scene_config
dict | tolerance
float64 50
50
| prompt_setting
stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (786, 742), initial velocity (-850.18, 702.49) m/s\n\n### Physics\n- constant gravity vector (9999.30, -118.66) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[751.1, 811.65]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[796.87, 833.45]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[853.83, 735.11]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[743.88, 746.37]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[733.93, 756.44]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[823.97, 765.33]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[808.42, 687.88]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[755.85, 680.39]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[770.53, 750.78]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[828.74, 796.58]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[745.85, 741.34]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[762.95, 684.91]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[848.5, 702.57]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[808.38, 821.42]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[744.67, 838.78]]) <= 50.0"
] |
ballsim
|
D_4_47
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
786,
742
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
-850.18,
702.49
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 89.32010653210423
},
"index": 47,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 2493131
},
"physics": {
"gravity_enabled": true,
"gravity_x": 9999.3,
"gravity_y": -118.66
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (775, 772), initial velocity (310.15, 965.27) m/s\n\n### Physics\n- constant gravity vector (5520.63, -8338.03) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[833.69, 826.73]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[861.82, 734.4]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[764.85, 659.91]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[723.4, 770.05]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[737.15, 796.8]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[806.11, 740.18]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[771.66, 651.71]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[647.66, 681.27]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[714.18, 728.11]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[845.42, 694.66]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[717.73, 679.89]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[705.91, 717.3]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[808.56, 753.34]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[804.39, 758.91]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[707.2, 729.26]]) <= 50.0"
] |
ballsim
|
D_4_1018
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
775,
772
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
310.15,
965.27
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 33.5086408709568
},
"index": 1018,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 4396101
},
"physics": {
"gravity_enabled": true,
"gravity_x": 5520.63,
"gravity_y": -8338.03
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (781, 751), initial velocity (-911.92, -584.96) m/s\n\n### Physics\n- constant gravity vector (9994.86, 320.62) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[739.91, 694.11]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[797.8, 642.75]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[871.5, 759.35]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[783.73, 829.07]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[740.09, 767.25]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[796.39, 708.63]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[785.37, 722.52]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[746.8, 792.41]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[800.4, 846.72]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[860.61, 731.62]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[744.51, 734.16]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[728.36, 739.9]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[811.83, 748.82]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[817.59, 687.35]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[751.53, 673.5]]) <= 50.0"
] |
ballsim
|
D_4_288
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
781,
751
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
-911.92,
-584.96
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 91.8373400530304
},
"index": 288,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 2034919
},
"physics": {
"gravity_enabled": true,
"gravity_x": 9994.86,
"gravity_y": 320.62
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (774, 770), initial velocity (1069.45, -239.08) m/s\n\n### Physics\n- constant gravity vector (-9937.98, -1111.97) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[831.13, 740.52]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[788.88, 699.92]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[669.9, 679.89]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[687.44, 833.17]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[776.39, 821.96]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[765.96, 722.82]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[704.94, 679.71]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[697.0, 835.32]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[753.26, 754.68]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[737.44, 629.38]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[623.17, 758.12]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[779.24, 755.61]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[835.92, 742.04]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[793.48, 717.32]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[651.66, 681.48]]) <= 50.0"
] |
ballsim
|
D_4_617
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
774,
770
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
1069.45,
-239.08
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -83.61567018076397
},
"index": 617,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 9131649
},
"physics": {
"gravity_enabled": true,
"gravity_x": -9937.98,
"gravity_y": -1111.97
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (773, 714), initial velocity (-620.54, -878.73) m/s\n\n### Physics\n- constant gravity vector (1672.85, -9859.09) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[736.82, 675.89]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[756.33, 760.15]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[792.68, 745.65]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[818.86, 678.93]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[733.5, 735.81]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[664.87, 694.09]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[755.69, 673.55]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[812.91, 723.04]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[826.01, 779.33]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[855.83, 737.03]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[789.68, 637.16]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[668.66, 679.24]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[679.57, 752.25]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[707.21, 726.97]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[759.28, 646.78]]) <= 50.0"
] |
ballsim
|
D_4_975
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
773,
714
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
-620.54,
-878.73
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 9.629981180462828
},
"index": 975,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 4992179
},
"physics": {
"gravity_enabled": true,
"gravity_x": 1672.85,
"gravity_y": -9859.09
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (774, 788), initial velocity (-212.79, 1071.0) m/s\n\n### Physics\n- constant gravity vector (5147.90, -8573.16) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[778.52, 852.13]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[834.53, 830.52]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[858.87, 692.82]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[701.03, 656.6]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[636.68, 736.18]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[701.13, 765.47]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[817.06, 709.04]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[722.89, 662.07]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[769.5, 711.99]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[834.49, 772.06]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[748.1, 820.23]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[713.18, 782.67]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[729.64, 659.79]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[845.81, 724.59]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[826.78, 836.69]]) <= 50.0"
] |
ballsim
|
D_4_583
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
774,
788
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
-212.79,
1071
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 30.983411559723486
},
"index": 583,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 2897632
},
"physics": {
"gravity_enabled": true,
"gravity_x": 5147.9,
"gravity_y": -8573.16
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (785, 786), initial velocity (311.11, 1134.04) m/s\n\n### Physics\n- constant gravity vector (9323.40, -3615.84) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[828.21, 797.66]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[796.44, 685.18]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[735.16, 668.43]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[825.5, 755.81]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[780.84, 712.5]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[759.32, 650.75]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[851.31, 750.63]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[806.91, 801.46]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[794.41, 668.08]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[770.17, 751.96]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[818.26, 850.17]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[853.77, 718.13]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[728.92, 654.45]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[760.35, 707.71]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[853.34, 745.9]]) <= 50.0"
] |
ballsim
|
D_4_803
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
785,
786
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
311.11,
1134.04
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 68.80251608817467
},
"index": 803,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 6708185
},
"physics": {
"gravity_enabled": true,
"gravity_x": 9323.4,
"gravity_y": -3615.84
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (788, 746), initial velocity (473.26, -892.92) m/s\n\n### Physics\n- constant gravity vector (2366.11, -9716.04) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[800.6, 688.7]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[712.64, 749.4]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[648.2, 712.73]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[735.67, 686.31]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[804.72, 713.89]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[809.58, 796.46]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[838.11, 781.86]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[826.54, 693.31]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[748.48, 691.71]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[726.42, 776.43]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[728.03, 763.98]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[753.29, 654.38]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[849.34, 716.33]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[830.46, 792.77]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[801.8, 784.22]]) <= 50.0"
] |
ballsim
|
D_4_537
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
788,
746
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
473.26,
-892.92
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 13.686633681384308
},
"index": 537,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 9366664
},
"physics": {
"gravity_enabled": true,
"gravity_x": 2366.11,
"gravity_y": -9716.04
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (747, 718), initial velocity (934.89, 465.94) m/s\n\n### Physics\n- constant gravity vector (8473.09, -5311.01) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[870.66, 741.97]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[772.03, 812.58]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[758.21, 830.07]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[829.26, 794.33]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[839.72, 671.08]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[708.31, 682.36]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[682.69, 669.52]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[791.17, 671.51]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[871.21, 776.14]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[769.02, 819.05]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[751.05, 808.68]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[817.8, 745.19]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[771.74, 692.99]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[702.27, 649.3]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[819.8, 693.25]]) <= 50.0"
] |
ballsim
|
D_4_700
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
747,
718
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
934.89,
465.94
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 57.92012997954941
},
"index": 700,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 3465800
},
"physics": {
"gravity_enabled": true,
"gravity_x": 8473.09,
"gravity_y": -5311.01
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (741, 742), initial velocity (-969.95, -266.93) m/s\n\n### Physics\n- constant gravity vector (8441.15, -5361.63) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[686.32, 688.43]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[752.45, 669.13]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[799.06, 724.39]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[739.56, 804.73]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[764.47, 831.46]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[845.67, 792.93]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[813.59, 674.25]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[749.51, 782.99]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[769.85, 838.12]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[844.0, 808.08]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[805.63, 673.82]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[694.57, 662.74]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[658.74, 704.83]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[707.13, 693.48]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[819.94, 676.76]]) <= 50.0"
] |
ballsim
|
D_4_957
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
741,
742
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
-969.95,
-266.93
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 57.57719474899707
},
"index": 957,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 8197305
},
"physics": {
"gravity_enabled": true,
"gravity_x": 8441.15,
"gravity_y": -5361.63
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (753, 742), initial velocity (-921.66, -522.59) m/s\n\n### Physics\n- constant gravity vector (-9245.92, 3809.58) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[647.93, 719.7]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[744.79, 831.15]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[749.08, 784.34]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[660.79, 687.36]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[749.06, 715.89]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[782.93, 794.89]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[724.34, 838.57]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[646.56, 720.05]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[734.51, 679.72]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[777.53, 742.88]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[728.08, 844.14]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[745.16, 764.67]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[712.93, 664.4]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[688.38, 739.72]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[702.72, 780.79]]) <= 50.0"
] |
ballsim
|
D_4_940
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
753,
742
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
-921.66,
-522.59
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -112.39304255544894
},
"index": 940,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 5653264
},
"physics": {
"gravity_enabled": true,
"gravity_x": -9245.92,
"gravity_y": 3809.58
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (733, 784), initial velocity (927.44, -629.74) m/s\n\n### Physics\n- constant gravity vector (348.02, -9993.94) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[827.49, 670.93]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[788.58, 788.38]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[740.25, 836.72]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[695.53, 784.87]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[673.51, 679.48]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[780.85, 778.03]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[849.87, 759.34]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[758.69, 668.71]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[687.82, 731.51]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[810.42, 773.06]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[848.94, 685.04]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[684.15, 683.12]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[771.88, 732.96]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[822.97, 750.1]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[690.02, 744.97]]) <= 50.0"
] |
ballsim
|
D_4_800
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
733,
784
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
927.44,
-629.74
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 1.9944053551656626
},
"index": 800,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 1805524
},
"physics": {
"gravity_enabled": true,
"gravity_x": 348.02,
"gravity_y": -9993.94
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (730, 782), initial velocity (-1124.42, 104.26) m/s\n\n### Physics\n- constant gravity vector (1505.33, -9886.05) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[625.1, 742.87]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[761.16, 698.45]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[835.06, 752.43]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[816.66, 828.77]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[745.8, 778.28]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[701.08, 655.71]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[810.2, 769.32]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[810.73, 732.85]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[774.99, 675.68]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[857.8, 789.42]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[744.08, 716.65]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[720.83, 727.12]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[787.21, 819.08]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[842.69, 801.72]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[815.85, 649.39]]) <= 50.0"
] |
ballsim
|
D_4_792
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
730,
782
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
-1124.42,
104.26
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 8.657795561737307
},
"index": 792,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 4323981
},
"physics": {
"gravity_enabled": true,
"gravity_x": 1505.33,
"gravity_y": -9886.05
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (740, 743), initial velocity (958.38, 294.48) m/s\n\n### Physics\n- constant gravity vector (4765.74, -8791.34) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[859.73, 728.38]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[747.83, 727.5]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[680.07, 662.35]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[832.92, 658.87]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[836.04, 767.77]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[738.6, 734.79]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[696.37, 656.73]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[748.42, 755.15]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[848.13, 765.66]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[815.66, 672.5]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[671.81, 668.5]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[750.92, 723.67]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[842.59, 722.0]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[721.23, 727.48]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[683.38, 675.13]]) <= 50.0"
] |
ballsim
|
D_4_325
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
740,
743
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
958.38,
294.48
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 28.46188086909928
},
"index": 325,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 5787308
},
"physics": {
"gravity_enabled": true,
"gravity_x": 4765.74,
"gravity_y": -8791.34
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (777, 767), initial velocity (706.51, 805.77) m/s\n\n### Physics\n- constant gravity vector (9825.96, 1857.53) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[806.33, 823.92]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[716.71, 820.36]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[725.49, 835.43]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[811.28, 832.09]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[832.51, 738.53]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[735.56, 742.31]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[736.88, 764.67]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[836.45, 805.6]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[757.39, 764.33]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[733.78, 726.05]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[808.42, 706.34]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[804.25, 769.66]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[769.05, 858.47]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[731.24, 791.7]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[791.69, 743.51]]) <= 50.0"
] |
ballsim
|
D_4_580
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
777,
767
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
706.51,
805.77
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 100.70506239652711
},
"index": 580,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 4860205
},
"physics": {
"gravity_enabled": true,
"gravity_x": 9825.96,
"gravity_y": 1857.53
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (772, 717), initial velocity (828.23, 686.55) m/s\n\n### Physics\n- constant gravity vector (-9902.04, 1396.30) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[805.19, 792.65]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[744.05, 870.95]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[679.2, 732.08]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[753.91, 706.35]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[824.38, 733.84]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[795.84, 775.29]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[668.27, 830.7]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[775.07, 761.98]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[808.62, 696.55]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[743.15, 645.08]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[632.02, 736.4]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[728.67, 844.27]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[822.5, 796.17]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[817.66, 761.89]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[713.8, 741.57]]) <= 50.0"
] |
ballsim
|
D_4_967
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
772,
717
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
828.23,
686.55
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -98.02643770296828
},
"index": 967,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 9658187
},
"physics": {
"gravity_enabled": true,
"gravity_x": -9902.04,
"gravity_y": 1396.3
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (712, 727), initial velocity (-805.41, -616.11) m/s\n\n### Physics\n- constant gravity vector (-3708.01, -9287.13) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[711.45, 659.67]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[781.66, 700.4]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[770.79, 754.33]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[722.82, 715.13]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[690.76, 711.01]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[692.38, 784.83]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[656.91, 765.77]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[694.94, 699.65]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[740.7, 667.86]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[664.74, 747.53]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[660.43, 689.3]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[769.38, 688.67]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[821.08, 671.39]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[706.84, 661.04]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[638.39, 757.89]]) <= 50.0"
] |
ballsim
|
D_4_576
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
712,
727
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
-805.41,
-616.11
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -21.764994931747843
},
"index": 576,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 7385592
},
"physics": {
"gravity_enabled": true,
"gravity_x": -3708.01,
"gravity_y": -9287.13
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (740, 780), initial velocity (-950.43, 318.97) m/s\n\n### Physics\n- constant gravity vector (9992.29, -392.51) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[695.04, 809.93]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[750.01, 835.93]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[847.01, 777.69]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[772.27, 678.65]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[713.35, 649.36]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[754.36, 722.61]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[844.75, 775.51]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[725.89, 724.03]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[706.96, 668.63]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[787.94, 643.14]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[863.61, 775.03]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[762.79, 867.92]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[700.96, 830.16]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[738.77, 788.38]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[876.5, 742.68]]) <= 50.0"
] |
ballsim
|
D_4_988
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
740,
780
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
-950.43,
318.97
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 87.75052040175358
},
"index": 988,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 1894039
},
"physics": {
"gravity_enabled": true,
"gravity_x": 9992.29,
"gravity_y": -392.51
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (767, 758), initial velocity (-1017.03, -151.29) m/s\n\n### Physics\n- constant gravity vector (-3751.10, -9269.80) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[646.5, 696.41]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[752.58, 763.72]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[832.53, 747.6]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[811.02, 661.89]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[707.68, 717.44]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[669.82, 779.1]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[768.59, 748.06]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[803.32, 670.27]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[668.75, 728.06]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[740.42, 693.14]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[772.44, 710.82]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[682.29, 782.44]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[692.02, 761.54]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[778.79, 647.99]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[668.03, 718.92]]) <= 50.0"
] |
ballsim
|
D_4_841
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
767,
758
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
-1017.03,
-151.29
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -22.03109229984824
},
"index": 841,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 9700036
},
"physics": {
"gravity_enabled": true,
"gravity_x": -3751.1,
"gravity_y": -9269.8
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (789, 733), initial velocity (-1076.75, -253.95) m/s\n\n### Physics\n- constant gravity vector (-9408.25, 3388.92) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[634.17, 724.59]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[740.12, 865.38]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[760.8, 733.47]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[709.05, 650.67]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[711.71, 806.14]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[775.87, 781.36]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[815.04, 695.39]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[760.12, 643.3]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[676.58, 715.19]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[731.29, 835.88]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[830.97, 813.82]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[836.56, 746.71]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[748.41, 713.53]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[679.06, 750.82]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[759.77, 845.58]]) <= 50.0"
] |
ballsim
|
D_4_370
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
789,
733
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
-1076.75,
-253.95
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -109.80937424477874
},
"index": 370,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 5138606
},
"physics": {
"gravity_enabled": true,
"gravity_x": -9408.25,
"gravity_y": 3388.92
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (728, 753), initial velocity (133.39, 1154.02) m/s\n\n### Physics\n- constant gravity vector (9210.09, -3895.40) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[787.5, 848.88]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[864.45, 725.55]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[715.42, 665.85]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[715.61, 727.16]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[829.16, 800.85]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[755.33, 719.81]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[760.73, 671.26]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[854.98, 786.52]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[751.58, 814.01]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[696.69, 741.16]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[733.9, 629.36]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[836.34, 732.33]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[808.91, 794.37]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[787.02, 647.62]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[727.86, 774.27]]) <= 50.0"
] |
ballsim
|
D_4_28
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
728,
753
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
133.39,
1154.02
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 67.07409017302471
},
"index": 28,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 2472993
},
"physics": {
"gravity_enabled": true,
"gravity_x": 9210.09,
"gravity_y": -3895.4
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (788, 738), initial velocity (-204.38, 1092.33) m/s\n\n### Physics\n- constant gravity vector (-9894.23, 1450.59) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[717.97, 854.5]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[665.47, 704.31]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[776.06, 652.29]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[811.9, 689.01]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[736.13, 770.82]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[727.02, 798.54]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[830.7, 753.07]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[835.43, 722.1]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[741.22, 705.64]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[690.22, 762.56]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[770.83, 851.68]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[828.58, 771.61]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[787.62, 706.2]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[655.06, 668.35]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[716.62, 795.72]]) <= 50.0"
] |
ballsim
|
D_4_727
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
788,
738
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
-204.38,
1092.33
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -98.34072288346651
},
"index": 727,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 6001636
},
"physics": {
"gravity_enabled": true,
"gravity_x": -9894.23,
"gravity_y": 1450.59
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (780, 763), initial velocity (-303.32, 1015.77) m/s\n\n### Physics\n- constant gravity vector (-9994.09, 343.73) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[700.71, 864.73]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[709.87, 710.6]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[735.25, 719.9]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[745.3, 849.34]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[655.42, 766.0]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[755.85, 733.07]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[831.75, 728.09]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[807.71, 726.54]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[683.74, 728.43]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[716.72, 817.23]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[758.38, 802.22]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[700.1, 678.53]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[703.25, 779.99]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[750.35, 806.85]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[789.27, 710.82]]) <= 50.0"
] |
ballsim
|
D_4_38
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
780,
763
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
-303.32,
1015.77
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -91.96980182714715
},
"index": 38,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 4518135
},
"physics": {
"gravity_enabled": true,
"gravity_x": -9994.09,
"gravity_y": 343.73
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (784, 736), initial velocity (556.92, -884.06) m/s\n\n### Physics\n- constant gravity vector (8896.73, 4565.98) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[832.42, 687.33]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[761.34, 752.06]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[779.39, 862.78]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[865.85, 760.15]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[811.21, 694.7]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[767.34, 782.54]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[760.94, 845.16]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[714.19, 775.45]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[756.41, 751.41]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[860.93, 764.36]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[752.42, 724.84]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[732.89, 730.99]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[802.04, 782.61]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[770.15, 818.2]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[688.71, 854.45]]) <= 50.0"
] |
ballsim
|
D_4_984
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
784,
736
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
556.92,
-884.06
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 117.16781201356586
},
"index": 984,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 5276164
},
"physics": {
"gravity_enabled": true,
"gravity_x": 8896.73,
"gravity_y": 4565.98
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (710, 755), initial velocity (992.72, 143.13) m/s\n\n### Physics\n- constant gravity vector (9329.43, 3600.24) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[856.04, 787.36]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[732.06, 723.49]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[682.86, 688.84]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[727.19, 690.32]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[864.82, 727.81]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[772.25, 834.81]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[659.25, 807.38]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[661.02, 797.95]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[771.83, 811.31]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[764.17, 783.59]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[679.48, 729.89]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[688.09, 712.18]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[789.63, 730.39]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[799.6, 717.27]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[705.65, 668.37]]) <= 50.0"
] |
ballsim
|
D_4_955
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
710,
755
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
992.72,
143.13
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 111.10169360675786
},
"index": 955,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 7783601
},
"physics": {
"gravity_enabled": true,
"gravity_x": 9329.43,
"gravity_y": 3600.24
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (754, 767), initial velocity (-808.24, -850.3) m/s\n\n### Physics\n- constant gravity vector (-9758.54, -2184.25) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[671.28, 690.5]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[800.43, 720.33]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[831.95, 728.31]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[765.68, 714.41]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[679.06, 710.65]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[774.27, 800.78]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[771.89, 869.07]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[665.1, 825.93]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[668.47, 660.91]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[795.19, 654.39]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[788.58, 738.2]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[684.38, 800.18]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[736.74, 735.23]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[761.41, 637.83]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[652.7, 730.26]]) <= 50.0"
] |
ballsim
|
D_4_509
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
754,
767
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
-808.24,
-850.3
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -77.38344457069617
},
"index": 509,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 3544457
},
"physics": {
"gravity_enabled": true,
"gravity_x": -9758.54,
"gravity_y": -2184.25
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (9 sides), radius 20.0m, initial position (731, 734), initial velocity (-844.02, -599.35) m/s\n\n### Physics\n- constant gravity vector (3311.23, -9435.88) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[694.91, 670.41]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[830.03, 702.55]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[755.42, 719.11]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[657.92, 674.17]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[741.98, 739.19]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[859.16, 709.86]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[731.16, 676.59]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[739.2, 714.85]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[846.13, 749.28]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[790.99, 752.77]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[704.56, 682.83]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[778.4, 693.55]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[804.85, 724.23]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[695.41, 715.66]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[688.9, 708.83]]) <= 50.0"
] |
ballsim
|
D_4_732
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
731,
734
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 9,
"velocity": [
-844.02,
-599.35
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 19.336968213495485
},
"index": 732,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 7838954
},
"physics": {
"gravity_enabled": true,
"gravity_x": 3311.23,
"gravity_y": -9435.88
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (775, 741), initial velocity (-1032.01, -195.65) m/s\n\n### Physics\n- constant gravity vector (-2440.23, -9697.69) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[659.57, 672.83]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[800.3, 656.63]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[841.94, 761.02]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[859.05, 768.05]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[849.14, 679.18]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[700.28, 648.12]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[629.16, 767.61]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[697.62, 797.23]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[744.81, 728.57]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[725.11, 665.52]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[651.94, 755.77]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[700.77, 809.7]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[725.17, 766.87]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[727.5, 632.49]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[838.63, 722.73]]) <= 50.0"
] |
ballsim
|
D_4_405
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
775,
741
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
-1032.01,
-195.65
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -14.12411104352642
},
"index": 405,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 374870
},
"physics": {
"gravity_enabled": true,
"gravity_x": -2440.23,
"gravity_y": -9697.69
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (760, 745), initial velocity (-939.9, 693.78) m/s\n\n### Physics\n- constant gravity vector (-6592.14, -7519.56) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[632.97, 776.69]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[755.27, 733.18]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[789.42, 666.35]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[643.2, 722.32]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[771.1, 703.1]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[807.87, 702.04]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[702.74, 757.38]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[708.51, 737.53]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[808.94, 645.78]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[696.24, 733.81]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[718.42, 746.65]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[820.75, 684.29]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[758.92, 717.29]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[655.8, 727.75]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[793.98, 663.02]]) <= 50.0"
] |
ballsim
|
D_4_756
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
760,
745
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
-939.9,
693.78
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -41.2399284981752
},
"index": 756,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 2368862
},
"physics": {
"gravity_enabled": true,
"gravity_x": -6592.14,
"gravity_y": -7519.56
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 8 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (773, 765), initial velocity (-276.83, 1040.95) m/s\n\n### Physics\n- constant gravity vector (9674.99, -2528.76) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[793.81, 856.42]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[838.72, 747.14]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[758.69, 693.33]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[701.59, 645.03]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[782.58, 671.25]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[800.48, 748.24]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[786.9, 853.06]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[801.87, 767.92]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[795.48, 682.95]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[704.2, 647.92]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[748.27, 680.8]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[839.55, 726.5]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[791.8, 844.32]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[777.18, 784.18]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[845.24, 667.45]]) <= 50.0"
] |
ballsim
|
D_4_718
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
773,
765
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
-276.83,
1040.95
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 8
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 75.35223484125322
},
"index": 718,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 9671251
},
"physics": {
"gravity_enabled": true,
"gravity_x": 9674.99,
"gravity_y": -2528.76
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 10 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (8 sides), radius 20.0m, initial position (773, 789), initial velocity (-737.68, 807.55) m/s\n\n### Physics\n- constant gravity vector (1603.92, -9870.53) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[707.27, 820.28]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[657.58, 752.85]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[719.94, 628.98]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[822.17, 708.54]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[699.89, 770.83]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[635.49, 720.82]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[774.75, 644.12]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[791.4, 777.29]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[796.51, 849.7]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[817.67, 823.42]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[854.86, 698.42]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[705.17, 667.07]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[713.31, 742.94]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[845.22, 755.42]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[766.64, 742.8]]) <= 50.0"
] |
ballsim
|
D_4_396
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
773,
789
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 8,
"velocity": [
-737.68,
807.55
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 10
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": 9.229651113287138
},
"index": 396,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 9856811
},
"physics": {
"gravity_enabled": true,
"gravity_x": 1603.92,
"gravity_y": -9870.53
}
}
| 50
|
function
|
[
{
"content": "## Polygon Dynamics Prediction\nIn this task, you will implement a single function predict_position(t) that computes the 2D positions of all balls at an arbitrary future time t under idealized mechanics. The function parses the scene configuration (containers, balls, and physics/meta), reconstructs the motions, detects and handles boundary collisions with finite-size treatment, and returns a list where each element is the [x, y] position (rounded to 2 decimals) of a ball at time t. Each evaluation of t must be computed directly from initial conditions and scene mechanics with no hidden state or accumulation across calls. Rendering, animation, and explanatory text are out of scope; prefer closed-form reasoning and avoid coarse time-stepping except where narrowly required for collision resolution.\n\n### Mechanics (General)\n- Kinematics: Use closed-form equations under constant acceleration: x(t)=x0+vx0*t+0.5*ax*t^2, y(t)=y0+vy0*t+0.5*ay*t^2.\n- Collisions: Perfectly elastic. Reflect velocity using v' = v - 2·dot(v, n̂)·n̂, where n̂ is the inward unit normal at the contact.\n- Finite size: Use polygon–polygon contact. Derive regular shapes from ('sides','radius','center','rotation'); irregular convex polygon balls use provided vertices.\n- Geometry: Irregular convex polygons (if present) are simple (non self-intersecting). Ball finite size must be respected in all interactions.\n- Units: Positions in meters; time in seconds; angles in radians; velocities in m/s; accelerations in m/s^2.\n- Cartesian Axes: +X is right, +Y is up.\n\n### Constraints\n- Implement only predict_position(t); no other entry points will be called.\n- No global variables; no variables defined outside the function.\n- Do not import external libraries (except math); do not perform I/O; do not print; do not use randomness.\n- Numerical output must be round(value, 2); normalize -0.0 to 0.0.\n\n### Verification and output contract\n- Return a list of positions per ball for the provided t: [[x1,y1],[x2,y2],...].\n- Each call must be computed independently (no state carry-over between calls).\n- You should assume that the ball will hit the wall and bounce back, which will be verified in test cases. \n\n\n### Scene description\n#### Containers\n- Container 1: regular polygon with 9 sides, radius 150.00m, center at (750, 750); initial orientation 0.000 rad\n\n#### Objects\n- Ball 1: regular polygon (10 sides), radius 20.0m, initial position (711, 721), initial velocity (1003.47, 169.58) m/s\n\n### Physics\n- constant gravity vector (-9279.65, 3726.68) m/s^2.\n\n### Dynamics\n- Gravity varies over time.\n\n### Conventions for this scene\n- Containers are convex regular polygons (parameters: 'sides', 'radius', 'center'), unless otherwise specified.\n- Angle baseline: By default, the initial orientation is 0.000 rad, pointing to the first vertex along +X (standard Cartesian axes); positive angles rotate CCW about the container center.\n- Polygon vertices (if provided) are CCW and form a simple convex polygon.\n- Container 'radius' denotes the circumradius (meters).\n- For balls: irregular convex polygons rely on provided vertices (no radius mentioned); regular polygons may be derived from 'sides/radius/center/rotation'.\n- Containers are kinematic (infinite mass, prescribed motion); impacts do not alter container motion.\n\n### Task\n- Number of balls: 1\n- Your should think step by step and write python code.\n- The final output should be in the following format: \n[Your thinking steps here ...](optional)\n```python\n[Your Python code here]\n```\n- Define predict_position(t) returning a list of length n_balls; each element is [x_i, y_i] (rounded to 2 decimals) for Ball i at time t (seconds)\n\n### Output\n- Required format: function predict_position(t: float) -> [[x1,y1],[x2,y2],...]; coordinates as 2-decimal floats",
"role": "user"
}
] |
[
"assert check_distance(predict_position(0.10), [[764.83, 756.64]]) <= 50.0",
"assert check_distance(predict_position(0.20), [[725.87, 829.54]]) <= 50.0",
"assert check_distance(predict_position(0.30), [[635.87, 779.47]]) <= 50.0",
"assert check_distance(predict_position(0.40), [[715.21, 685.01]]) <= 50.0",
"assert check_distance(predict_position(0.50), [[704.47, 643.18]]) <= 50.0",
"assert check_distance(predict_position(0.60), [[641.1, 724.65]]) <= 50.0",
"assert check_distance(predict_position(0.70), [[711.14, 843.38]]) <= 50.0",
"assert check_distance(predict_position(0.80), [[728.95, 769.34]]) <= 50.0",
"assert check_distance(predict_position(0.90), [[662.0, 686.95]]) <= 50.0",
"assert check_distance(predict_position(1.00), [[685.02, 795.18]]) <= 50.0",
"assert check_distance(predict_position(1.10), [[760.55, 852.93]]) <= 50.0",
"assert check_distance(predict_position(1.20), [[786.14, 829.91]]) <= 50.0",
"assert check_distance(predict_position(1.30), [[693.72, 799.89]]) <= 50.0",
"assert check_distance(predict_position(1.40), [[635.53, 700.32]]) <= 50.0",
"assert check_distance(predict_position(1.50), [[738.71, 715.03]]) <= 50.0"
] |
ballsim
|
D_4_829
| 4
|
[
0.1,
0.2,
0.3,
0.4,
0.5,
0.6,
0.7,
0.8,
0.9,
1,
1.1,
1.2,
1.3,
1.4,
1.5
] |
{
"balls": [
{
"color": [
240,
180,
60
],
"density": 1,
"position": [
711,
721
],
"radius": 20,
"restitution": 1,
"rotation_affected": false,
"sides": 10,
"velocity": [
1003.47,
169.58
]
}
],
"boxes": [
{
"center": [
750,
750
],
"diameter": 300,
"friction": 0,
"restitution": 1,
"rotation": 0,
"rotation_speed": 0,
"sides": 9
}
],
"meta": {
"difficulty": "extreme",
"difficulty_config": {
"description": "Extreme: 极小空间，极复杂布局，极高密度，极高碰撞频率"
},
"difficulty_level": 4,
"generation_config": {
"multi_container_config": {
"min_gap": 50
}
},
"gravity_profile": {
"tilt_angle_deg": -111.88027378805604
},
"index": 829,
"scene_space": {
"center": [
750,
750
],
"height": 1500,
"units": "meters",
"width": 1500
},
"scene_type": "D",
"seed": 6085375
},
"physics": {
"gravity_enabled": true,
"gravity_x": -9279.65,
"gravity_y": 3726.68
}
}
| 50
|
function
|
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