id
stringlengths 19
37
| setting_key
stringclasses 7
values | setting_name
stringclasses 7
values | messages
listlengths 1
1
| ground_truth
stringlengths 1
6
| dataset
stringclasses 1
value | omega_config
stringclasses 7
values | omega_split
stringclasses 1
value |
|---|---|---|---|---|---|---|---|
comp_vertex_color_test_035
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular heptagon is independently colored with either green, yellow, or magenta with equal probability. The probability that the heptagon can then be rotated so that all of the magenta vertices end up at positions where there had been green vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3218
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_036
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular hexagon is independently colored with either red, yellow, or magenta with equal probability. The probability that the hexagon can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
368
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_037
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular diamond is independently colored with either red, blue, or green with equal probability. The probability that the diamond can then be rotated so that all of the green vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_038
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rectangle is colored either yellow with probability 1 or blue with probability 0. The probability that the rectangle can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_039
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular square is independently colored with either blue, green, or magenta with equal probability. The probability that the square can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_040
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular square is independently colored with either red, blue, or green with equal probability. The probability that the square can then be rotated so that all of the green vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_041
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular triangle is independently colored with either blue, green, yellow, or magenta with equal probability. The probability that the triangle can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
53
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_042
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular pentagon is independently colored with either red, green, or yellow with equal probability. The probability that the pentagon can then be rotated so that all of the yellow vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
380
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_043
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular hexagon is independently colored with either red, blue, or magenta with equal probability. The probability that the hexagon can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
368
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_044
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular diamond is independently colored with either red, blue, or magenta with equal probability. The probability that the diamond can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_045
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular diamond is colored either yellow with probability 0.25 or blue with probability 0.75. The probability that the diamond can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
323
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_046
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular diamond is independently colored with either blue, green, or magenta with equal probability. The probability that the diamond can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_047
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rectangle is independently colored with either blue, green, or magenta with equal probability. The probability that the rectangle can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_048
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular octagon is independently colored with either blue, green, or magenta with equal probability. The probability that the octagon can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
9403
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_049
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular octagon is independently colored with either blue, green, or yellow with equal probability. The probability that the octagon can then be rotated so that all of the yellow vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
9403
|
math
|
comp_vertex_color
|
test
|
comp_grid_chips_test_000
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 6 indistinguishable white chips and 6 indistinguishable black chips. Find the number of ways to place some of these chips in the 6 unit cells of a 1 x 6 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
64
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_001
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 28 indistinguishable white chips and 28 indistinguishable black chips. Find the number of ways to place some of these chips in the 28 unit cells of a 7 x 4 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1766
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_002
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 35 indistinguishable white chips and 35 indistinguishable black chips. Find the number of ways to place some of these chips in the 35 unit cells of a 5 x 7 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3782
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_003
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 52 indistinguishable white chips and 52 indistinguishable black chips. Find the number of ways to place some of these chips in the 52 unit cells of a 4 x 13 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
114662
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_004
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 16 indistinguishable white chips and 16 indistinguishable black chips. Find the number of ways to place some of these chips in the 16 unit cells of a 2 x 8 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
510
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_005
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 30 indistinguishable white chips and 30 indistinguishable black chips. Find the number of ways to place some of these chips in the 30 unit cells of a 3 x 10 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6134
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_006
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 77 indistinguishable white chips and 77 indistinguishable black chips. Find the number of ways to place some of these chips in the 77 unit cells of a 11 x 7 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
257798
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_007
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 72 indistinguishable white chips and 72 indistinguishable black chips. Find the number of ways to place some of these chips in the 72 unit cells of a 8 x 9 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
129542
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_008
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 48 indistinguishable white chips and 48 indistinguishable black chips. Find the number of ways to place some of these chips in the 48 unit cells of a 12 x 4 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
57318
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_009
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 35 indistinguishable white chips and 35 indistinguishable black chips. Find the number of ways to place some of these chips in the 35 unit cells of a 7 x 5 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3782
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_010
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 28 indistinguishable white chips and 28 indistinguishable black chips. Find the number of ways to place some of these chips in the 28 unit cells of a 4 x 7 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1766
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_011
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 11 indistinguishable white chips and 11 indistinguishable black chips. Find the number of ways to place some of these chips in the 11 unit cells of a 1 x 11 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2048
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_012
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 52 indistinguishable white chips and 52 indistinguishable black chips. Find the number of ways to place some of these chips in the 52 unit cells of a 13 x 4 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
114662
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_013
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 8 indistinguishable white chips and 8 indistinguishable black chips. Find the number of ways to place some of these chips in the 8 unit cells of a 4 x 2 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
30
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_014
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 4 indistinguishable white chips and 4 indistinguishable black chips. Find the number of ways to place some of these chips in the 4 unit cells of a 2 x 2 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_015
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 72 indistinguishable white chips and 72 indistinguishable black chips. Find the number of ways to place some of these chips in the 72 unit cells of a 12 x 6 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
253830
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_016
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 22 indistinguishable white chips and 22 indistinguishable black chips. Find the number of ways to place some of these chips in the 22 unit cells of a 2 x 11 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4094
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_017
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 42 indistinguishable white chips and 42 indistinguishable black chips. Find the number of ways to place some of these chips in the 42 unit cells of a 7 x 6 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7814
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_018
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 12 indistinguishable white chips and 12 indistinguishable black chips. Find the number of ways to place some of these chips in the 12 unit cells of a 1 x 12 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4096
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_019
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 60 indistinguishable white chips and 60 indistinguishable black chips. Find the number of ways to place some of these chips in the 60 unit cells of a 15 x 4 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
458726
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_020
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 30 indistinguishable white chips and 30 indistinguishable black chips. Find the number of ways to place some of these chips in the 30 unit cells of a 10 x 3 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6134
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_021
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 14 indistinguishable white chips and 14 indistinguishable black chips. Find the number of ways to place some of these chips in the 14 unit cells of a 2 x 7 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
254
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_022
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 2 indistinguishable white chips and 2 indistinguishable black chips. Find the number of ways to place some of these chips in the 2 unit cells of a 1 x 2 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_023
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 48 indistinguishable white chips and 48 indistinguishable black chips. Find the number of ways to place some of these chips in the 48 unit cells of a 8 x 6 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
15750
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_024
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 8 indistinguishable white chips and 8 indistinguishable black chips. Find the number of ways to place some of these chips in the 8 unit cells of a 2 x 4 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
30
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_025
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 40 indistinguishable white chips and 40 indistinguishable black chips. Find the number of ways to place some of these chips in the 40 unit cells of a 5 x 8 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7622
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_026
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 42 indistinguishable white chips and 42 indistinguishable black chips. Find the number of ways to place some of these chips in the 42 unit cells of a 6 x 7 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7814
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_027
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 78 indistinguishable white chips and 78 indistinguishable black chips. Find the number of ways to place some of these chips in the 78 unit cells of a 6 x 13 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
507782
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_028
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 45 indistinguishable white chips and 45 indistinguishable black chips. Find the number of ways to place some of these chips in the 45 unit cells of a 9 x 5 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
15302
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_029
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 16 indistinguishable white chips and 16 indistinguishable black chips. Find the number of ways to place some of these chips in the 16 unit cells of a 8 x 2 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
510
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_030
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 66 indistinguishable white chips and 66 indistinguishable black chips. Find the number of ways to place some of these chips in the 66 unit cells of a 11 x 6 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
126854
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_031
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 8 indistinguishable white chips and 8 indistinguishable black chips. Find the number of ways to place some of these chips in the 8 unit cells of a 1 x 8 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
256
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_032
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 27 indistinguishable white chips and 27 indistinguishable black chips. Find the number of ways to place some of these chips in the 27 unit cells of a 3 x 9 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3062
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_033
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 26 indistinguishable white chips and 26 indistinguishable black chips. Find the number of ways to place some of these chips in the 26 unit cells of a 2 x 13 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
16382
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_034
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 24 indistinguishable white chips and 24 indistinguishable black chips. Find the number of ways to place some of these chips in the 24 unit cells of a 6 x 4 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
870
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_035
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 20 indistinguishable white chips and 20 indistinguishable black chips. Find the number of ways to place some of these chips in the 20 unit cells of a 2 x 10 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2046
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_036
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 33 indistinguishable white chips and 33 indistinguishable black chips. Find the number of ways to place some of these chips in the 33 unit cells of a 3 x 11 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12278
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_037
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 18 indistinguishable white chips and 18 indistinguishable black chips. Find the number of ways to place some of these chips in the 18 unit cells of a 3 x 6 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
374
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_038
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 70 indistinguishable white chips and 70 indistinguishable black chips. Find the number of ways to place some of these chips in the 70 unit cells of a 5 x 14 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
491462
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_039
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 2 indistinguishable white chips and 2 indistinguishable black chips. Find the number of ways to place some of these chips in the 2 unit cells of a 2 x 1 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_040
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 54 indistinguishable white chips and 54 indistinguishable black chips. Find the number of ways to place some of these chips in the 54 unit cells of a 6 x 9 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
31622
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_041
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 64 indistinguishable white chips and 64 indistinguishable black chips. Find the number of ways to place some of these chips in the 64 unit cells of a 8 x 8 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
64518
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_042
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 72 indistinguishable white chips and 72 indistinguishable black chips. Find the number of ways to place some of these chips in the 72 unit cells of a 6 x 12 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
253830
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_043
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 77 indistinguishable white chips and 77 indistinguishable black chips. Find the number of ways to place some of these chips in the 77 unit cells of a 7 x 11 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
257798
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_044
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 88 indistinguishable white chips and 88 indistinguishable black chips. Find the number of ways to place some of these chips in the 88 unit cells of a 11 x 8 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
519686
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_045
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 11 indistinguishable white chips and 11 indistinguishable black chips. Find the number of ways to place some of these chips in the 11 unit cells of a 11 x 1 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2048
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_046
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 5 indistinguishable white chips and 5 indistinguishable black chips. Find the number of ways to place some of these chips in the 5 unit cells of a 5 x 1 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
32
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_047
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 3 indistinguishable white chips and 3 indistinguishable black chips. Find the number of ways to place some of these chips in the 3 unit cells of a 3 x 1 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_048
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 30 indistinguishable white chips and 30 indistinguishable black chips. Find the number of ways to place some of these chips in the 30 unit cells of a 2 x 15 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
65534
|
math
|
comp_grid_chips
|
test
|
comp_grid_chips_test_049
|
comp_grid_chips
|
Logic (Grid Chip) + Combinatorics (Probability)
|
[
{
"content": "There is a collection of 40 indistinguishable white chips and 40 indistinguishable black chips. Find the number of ways to place some of these chips in the 40 unit cells of a 8 x 5 grid such that: 1) Each cell contains at most one chip, 2) All chips in the same row and all chips in the same column have the same colour, and 3) Any additional chip placed on the grid would violate one or more of the previous two conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7622
|
math
|
comp_grid_chips
|
test
|
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