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comp_vertex_color_train_0000 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is blue. The 9-gon is then rotated clockwise... | 2 | math |
comp_vertex_color_train_0001 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue. The 6-gon is then rotated clockwise by 1 positions, so that the vertex that was originall... | 2 | math |
comp_vertex_color_train_0002 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is red. The 8-gon is then rotated clockwise by 6 positions, so... | 2 | math |
comp_vertex_color_train_0003 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is red, vertex 3 is uncolored, vertex 4 is red, vertex 5 is uncolored, vertex 6 is red, vertex 7 is blue, vertex 8 is blue. How many valid color... | 2 | math |
comp_vertex_color_train_0004 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy a... | 3 | math |
comp_vertex_color_train_0005 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is uncolored, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist... | 5 | math |
comp_vertex_color_train_0006 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is blue. What is the maximum number of blue vertices that can be made to occupy a position where ... | 2 | math |
comp_vertex_color_train_0007 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is blue. How many valid colo... | 0 | math |
comp_vertex_color_train_0008 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red. The 8-gon is then rotated clockwise by 6 positions, ... | 2 | math |
comp_vertex_color_train_0009 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is uncolored. How many valid colorings of the uncolored vertices exist such that the 6-... | 2 | math |
comp_vertex_color_train_0010 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue. What is the maximum number of blue vertices that can be made to occupy a position where th... | 2 | math |
comp_vertex_color_train_0011 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is red, vertex 5 is blue, vertex 6 is red. How many valid colorings of the uncolored vertices exis... | 3 | math |
comp_vertex_color_train_0012 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is blue. The 8-gon is then rotated clockwise by 3 positions, so... | 2 | math |
comp_vertex_color_train_0013 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist... | 5 | math |
comp_vertex_color_train_0014 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue. What is the maximum number of blue vertices that can... | 2 | math |
comp_vertex_color_train_0015 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue. What is the maximum number of blue vertices that can b... | 3 | math |
comp_vertex_color_train_0016 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue. What is the maximum number of red ... | 2 | math |
comp_vertex_color_train_0017 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is uncolored, vertex 3 is red, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is red, vertex 7 is red, vertex 8 is uncolored. How many valid colori... | 0 | math |
comp_vertex_color_train_0018 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is red, vertex 5 is blue. How many valid colorings of the uncolored vertices exist such that the 6... | 2 | math |
comp_vertex_color_train_0019 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue. What is the maximum number of red vertices that can be made to occupy a position where the... | 2 | math |
comp_vertex_color_train_0020 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is red. How many valid colorings of the uncolored vertices exis... | 2 | math |
comp_vertex_color_train_0021 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy ... | 3 | math |
comp_vertex_color_train_0022 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is uncolored, vertex 5 is red, vertex 6 is uncolored, vertex 7 is blue. How many valid colorings of the uncolor... | 0 | math |
comp_vertex_color_train_0023 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is blue. What is the maximum number of blue ver... | 3 | math |
comp_vertex_color_train_0024 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is red. The 9-gon is then rotated clockwise... | 3 | math |
comp_vertex_color_train_0025 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is red. What is the maximum number of blue ve... | 3 | math |
comp_vertex_color_train_0026 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is red. What is the maximum number of blue vertices that can be... | 3 | math |
comp_vertex_color_train_0027 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is red. How many valid colorings of the uncolored vertices exist such that the 6... | 2 | math |
comp_vertex_color_train_0028 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue. How many valid colorings of the unco... | 8 | math |
comp_vertex_color_train_0029 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is red, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist ... | 7 | math |
comp_vertex_color_train_0030 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is red. What is the maximum number of blue v... | 3 | math |
comp_vertex_color_train_0031 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue. What is the maximum number of red vertices that can... | 1 | math |
comp_vertex_color_train_0032 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red. What is the maximum number of blue vertices that can ... | 3 | math |
comp_vertex_color_train_0033 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue. What is the maximum number of red vertices that can be made to occupy ... | 2 | math |
comp_vertex_color_train_0034 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue. What is the maximum number of red v... | 3 | math |
comp_vertex_color_train_0035 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be ... | 3 | math |
comp_vertex_color_train_0036 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue. The 8-gon is then rotated clockwise by 6 positions, s... | 2 | math |
comp_vertex_color_train_0037 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue. The 7-gon is then rotated clockwise by 2 positions, so that the vertex ... | 2 | math |
comp_vertex_color_train_0038 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue. What is the maximum number of red vertices that can be made to occupy a position where the... | 2 | math |
comp_vertex_color_train_0039 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red. What is the maximum number of red vertices that can be m... | 2 | math |
comp_vertex_color_train_0040 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is blue. What is the maximum number of blue vertices that can be made to occupy a position where... | 1 | math |
comp_vertex_color_train_0041 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy a ... | 2 | math |
comp_vertex_color_train_0042 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is blue, vertex 2 is red, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolor... | 7 | math |
comp_vertex_color_train_0043 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy a p... | 1 | math |
comp_vertex_color_train_0044 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is blue, vertex 6 is red. How many valid colorings of the uncolored vertices exis... | 3 | math |
comp_vertex_color_train_0045 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue. The 9-gon is then rotated clockwise b... | 2 | math |
comp_vertex_color_train_0046 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is blue, vertex 7 is red. How many valid colorings of the uncol... | 1 | math |
comp_vertex_color_train_0047 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red. How many valid colorings of the uncolo... | 5 | math |
comp_vertex_color_train_0048 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy... | 2 | math |
comp_vertex_color_train_0049 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is uncolored. How many valid colorings of the uncolored vertices exist such that the 6... | 3 | math |
comp_vertex_color_train_0050 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red. What is the maximum number of red ... | 2 | math |
comp_vertex_color_train_0051 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red. What is the maximum number of red vertices that can be made to occupy a position where the... | 2 | math |
comp_vertex_color_train_0052 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue. What is the maximum number of blue v... | 3 | math |
comp_vertex_color_train_0053 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is red, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is uncolored. How many valid colorin... | 0 | math |
comp_vertex_color_train_0054 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be... | 4 | math |
comp_vertex_color_train_0055 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue. What is the maximum number of blue vertices that can be made to occupy a position where the... | 1 | math |
comp_vertex_color_train_0056 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue. What is the maximum number of red vertices that can be made to occupy a... | 2 | math |
comp_vertex_color_train_0057 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy a... | 2 | math |
comp_vertex_color_train_0058 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue. What is the maximum number of blue vertices that can be made to occupy a position where ... | 2 | math |
comp_vertex_color_train_0059 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is uncolored, vertex 4 is red, vertex 5 is uncolored, vertex 6 is uncolored, vertex 7 is blue. How many valid colorings of the uncolo... | 5 | math |
comp_vertex_color_train_0060 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is uncolored, vertex 7 is uncolored. How many valid colorings of the uncol... | 6 | math |
comp_vertex_color_train_0061 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is uncolored, vertex 6 is red. How many valid colorings of the uncolored vertices exi... | 7 | math |
comp_vertex_color_train_0062 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is red. The 7-gon is then rotated clockwise by 1 positions, so that the vertex... | 1 | math |
comp_vertex_color_train_0063 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is uncolored, vertex 8 is red. How many valid colori... | 0 | math |
comp_vertex_color_train_0064 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red. How many valid colorings of the uncolo... | 1 | math |
comp_vertex_color_train_0065 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is uncolored, vertex 7 is blue. How many valid colorings of the uncolo... | 2 | math |
comp_vertex_color_train_0066 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red. How many valid colorings of the uncolored vertices exist such that the 6-... | 3 | math |
comp_vertex_color_train_0067 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is blue. How many valid colorings of the uncolored vertices exist such that the 6-... | 5 | math |
comp_vertex_color_train_0068 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is blue. The 9-gon is then rotated clockwis... | 2 | math |
comp_vertex_color_train_0069 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be... | 4 | math |
comp_vertex_color_train_0070 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red. What is the maximum number of red vertices that can be made to occupy a position where t... | 2 | math |
comp_vertex_color_train_0071 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy a... | 3 | math |
comp_vertex_color_train_0072 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be m... | 2 | math |
comp_vertex_color_train_0073 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue. The 6-gon is then rotated clockwise by 5 positions, so that the vertex that was originally... | 2 | math |
comp_vertex_color_train_0074 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red. What is the maximum number of red vertices that can be made to occupy a ... | 2 | math |
comp_vertex_color_train_0075 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exis... | 3 | math |
comp_vertex_color_train_0076 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue. What is the maximum number of blue vertices that can be made to occupy... | 2 | math |
comp_vertex_color_train_0077 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue. The 8-gon is then rotated clockwise by 2 positions, ... | 2 | math |
comp_vertex_color_train_0078 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red. What is the maximum number of red vertices that can be made to occupy a position where t... | 2 | math |
comp_vertex_color_train_0079 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red. What is the maximum number of blue vertices that can be... | 3 | math |
comp_vertex_color_train_0080 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is blue. The 9-gon is then rotated clockwise ... | 3 | math |
comp_vertex_color_train_0081 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue. What is the maximum number of red ve... | 3 | math |
comp_vertex_color_train_0082 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue. The 7-gon is then rotated clockwise by 2 positions, so that the vertex ... | 1 | math |
comp_vertex_color_train_0083 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy... | 2 | math |
comp_vertex_color_train_0084 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue. What is the maximum number of blue vertices that can be made to occupy a position where t... | 3 | math |
comp_vertex_color_train_0085 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is red, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is red, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exis... | 3 | math |
comp_vertex_color_train_0086 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolor... | 7 | math |
comp_vertex_color_train_0087 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red. What is the maximum number of red vertices that can be made to occupy... | 1 | math |
comp_vertex_color_train_0088 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolo... | 3 | math |
comp_vertex_color_train_0089 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red. The 6-gon is then rotated clockwise by 4 positions, so that the vertex that was originally... | 2 | math |
comp_vertex_color_train_0090 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red. What is the maximum number of blue vertices that can b... | 4 | math |
comp_vertex_color_train_0091 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red. What is the maximum number of blue vertices that can be made to occupy a position where th... | 2 | math |
comp_vertex_color_train_0092 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is uncolored, vertex 8 is uncolored. How many valid color... | 2 | math |
comp_vertex_color_train_0093 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is red. What is the maximum number of blue vertices that can be... | 3 | math |
comp_vertex_color_train_0094 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colo... | 4 | math |
comp_vertex_color_train_0095 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be ma... | 1 | math |
comp_vertex_color_train_0096 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is red, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is blue. How many valid colorings of the uncolored vertices exis... | 3 | math |
comp_vertex_color_train_0097 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 8-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue. How many valid colorings of the uncol... | 3 | math |
comp_vertex_color_train_0098 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue. The 7-gon is then rotated clockwise by 5 positions, so that the vertex t... | 2 | math |
comp_vertex_color_train_0099 | comp_vertex_color | Geometry (Polygon Color) + Combinatorics (Probability) | [
{
"role": "user",
"content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red. What is the maximum number of red vertices that can be made to occupy a position where the... | 2 | math |
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