messages listlengths 1 1 | ground_truth stringclasses 9 values | dataset stringclasses 1 value |
|---|---|---|
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{
"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 uncolored, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored, vertex 8 is red. How many valid colorings of the uncolored ... | 7 | MATH |
[
{
"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 red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red. What is the maximum number of red vertices that can be ma... | 3 | MATH |
[
{
"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 uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is red. How many valid colorings of the uncolore... | 1 | MATH |
[
{
"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 uncolored, 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 uncolored vertices exist s... | 6 | MATH |
[
{
"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 uncolored, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red. How many valid colorings of the uncolored vertices exist ... | 6 | MATH |
[
{
"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 red, vertex 7 is red. The 8-gon is then rotated clockwise by 4 positions, so that the vertex tha... | 2 | MATH |
[
{
"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 red, vertex 5 is blue, vertex 6 is red, vertex 7 is red. What is the maximum number of blue vertices that can be made to occupy a p... | 4 | MATH |
[
{
"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 red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue. What is the maximum number of blue vertices that can be ... | 4 | MATH |
[
{
"content": "A 9-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 blue, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is red, vertex 8 is red. How many valid colorings of the uncolored... | 0 | MATH |
[
{
"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 blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist... | 7 | MATH |
[
{
"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 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 be made to occupy a ... | 3 | MATH |
[
{
"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 red, 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 made to occupy a pos... | 2 | MATH |
[
{
"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 uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored, vertex 8 is red. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"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 red, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is blue, vertex 8 is red. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"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 red, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is blue. What is the maximum number of blue vertices that can be ... | 3 | MATH |
[
{
"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 uncolored, vertex 6 is blue, vertex 7 is red, vertex 8 is red. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"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 blue vertices that can be made to occupy a p... | 4 | MATH |
[
{
"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 uncolored, vertex 6 is red, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colorings of the uncolored... | 4 | MATH |
[
{
"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 blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a pos... | 3 | MATH |
[
{
"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 red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is blue. The 9-gon is then rotated clockwise by 5 positions, so that ... | 1 | MATH |
[
{
"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 blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue. The 9-gon is then rotated clockwise by 5 positions, so ... | 2 | MATH |
[
{
"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 red, vertex 5 is blue, vertex 6 is red, vertex 7 is blue. The 8-gon is then rotated clockwise by 2 positions, so that the vertex tha... | 2 | MATH |
[
{
"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 red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"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 blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is red. How many valid colorings of the uncolore... | 1 | MATH |
[
{
"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 blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue. What is the maximum number of blue vertices that can b... | 2 | MATH |
[
{
"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 blue, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a p... | 3 | MATH |
[
{
"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 red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is red. The 8-gon is then rotated clockwise by 4 positions, so that the vertex that ... | 0 | MATH |
[
{
"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 uncolored, vertex 3 is red, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue, vertex 8 is red. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"content": "A 9-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 red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored, vertex 8 is blue. How many valid colorings of the uncolored... | 1 | MATH |
[
{
"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 blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is uncolored, vertex 7 is blue. How many valid colorings of the uncolored vertices exist ... | 2 | MATH |
[
{
"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 red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue. What is the maximum number of red vertices that can be m... | 3 | MATH |
[
{
"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 blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue. How many valid colorings of the uncolored vertices exist... | 0 | MATH |
[
{
"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 blue, vertex 7 is red, vertex 8 is red. What is the maximum number of blue vertices that can be ma... | 3 | MATH |
[
{
"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 red, vertex 6 is blue, vertex 7 is red. The 8-gon is then rotated clockwise by 4 positions, so that the vertex that... | 1 | MATH |
[
{
"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 red, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is blue. What is the maximum number of red vertices that can be m... | 3 | MATH |
[
{
"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 blue, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red. How many valid colorings of the uncolored vertices exist... | 0 | MATH |
[
{
"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 red, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue. How many valid colorings of the uncolor... | 0 | MATH |
[
{
"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 red, vertex 3 is red, 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 made to occupy a posi... | 2 | MATH |
[
{
"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 red, vertex 5 is red, vertex 6 is red, vertex 7 is red. What is the maximum number of blue vertices that can be made to occupy a p... | 4 | MATH |
[
{
"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 red, vertex 6 is blue, vertex 7 is red. The 8-gon is then rotated clockwise by 3 positions, so that the vertex that... | 2 | MATH |
[
{
"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 blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red. What is the maximum number of blue vertices that can be made to occupy a po... | 3 | MATH |
[
{
"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 red, vertex 7 is blue. The 8-gon is then rotated clockwise by 3 positions, so that the vertex that ... | 2 | MATH |
[
{
"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 red, vertex 6 is red, vertex 7 is red, vertex 8 is red. What is the maximum number of red vertices that can be mad... | 3 | MATH |
[
{
"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 made to occupy a posi... | 2 | MATH |
[
{
"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 red, vertex 6 is red, vertex 7 is red, vertex 8 is blue. What is the maximum number of red vertices that can be ma... | 4 | MATH |
[
{
"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 red, vertex 7 is red. The 8-gon is then rotated clockwise by 5 positions, so that the vertex that w... | 1 | MATH |
[
{
"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 red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is red. What is the maximum number of red vertices that can be made... | 2 | MATH |
[
{
"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 red, vertex 4 is uncolored, vertex 5 is uncolored, vertex 6 is red, vertex 7 is red, vertex 8 is red. How many valid colorings of the uncolored ... | 0 | MATH |
[
{
"content": "A 9-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 red, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is red, vertex 8 is blue. How many valid colorings of the uncolore... | 1 | MATH |
[
{
"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 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. What is the maximum number of red vertices that can be mad... | 3 | MATH |
[
{
"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 red, vertex 6 is blue, vertex 7 is red, vertex 8 is red. What is the maximum number of red vertices that can be ma... | 3 | MATH |
[
{
"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 red, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a posi... | 2 | MATH |
[
{
"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 red, vertex 7 is blue. What is the maximum number of red vertices that can be made to occupy a po... | 4 | MATH |
[
{
"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 uncolored, vertex 6 is uncolored, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist s... | 2 | MATH |
[
{
"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 red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is blue. What is the maximum number of blue vertices that can be made to occupy a posi... | 1 | MATH |
[
{
"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 uncolored, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue. How many valid colorings of the uncolored vertices exist ... | 5 | MATH |
[
{
"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 uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is blue. How many valid colorings of the uncolored vertices exist s... | 5 | MATH |
[
{
"content": "A 9-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 red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is red. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"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 red, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red. The 8-gon is then rotated clockwise by 3 positions, so that the vertex th... | 2 | MATH |
[
{
"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 red, vertex 7 is blue. The 8-gon is then rotated clockwise by 1 positions, so that the vertex tha... | 3 | MATH |
[
{
"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 red, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colorings of the uncolore... | 1 | MATH |
[
{
"content": "A 9-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 blue, vertex 5 is red, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colorings of the uncolored... | 0 | MATH |
[
{
"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 red, vertex 6 is red, vertex 7 is blue. The 8-gon is then rotated clockwise by 4 positions, so that the vertex tha... | 2 | MATH |
[
{
"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 blue, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is red. What is the maximum number of red vertices that can be m... | 3 | MATH |
[
{
"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 blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a po... | 3 | MATH |
[
{
"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 blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist su... | 8 | MATH |
[
{
"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 red, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is red. What is the maximum number of blue vertices that can be ma... | 3 | MATH |
[
{
"content": "A 9-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, vertex 6 is red, vertex 7 is blue, vertex 8 is red. How many valid colorings of the uncolored... | 0 | MATH |
[
{
"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 uncolored, vertex 4 is red, vertex 5 is uncolored, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist s... | 1 | MATH |
[
{
"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 blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red. The 8-gon is then rotated clockwise by 7 positions, so that the vertex th... | 2 | MATH |
[
{
"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 red, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red. What is the maximum number of red vertices that can be mad... | 3 | MATH |
[
{
"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 red, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist s... | 1 | MATH |
[
{
"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 red, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red. What is the maximum number of blue vertices that can be m... | 3 | MATH |
[
{
"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 blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue. The 8-gon is then rotated clockwise by 2 positions, so that the vertex tha... | 2 | MATH |
[
{
"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 blue, vertex 3 is red, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is blue. How many valid colorings of the uncolored vertices exist... | 7 | MATH |
[
{
"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 blue, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exis... | 8 | MATH |
[
{
"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 blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colorings of the uncolored... | 5 | MATH |
[
{
"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 red, vertex 6 is blue, vertex 7 is blue. The 8-gon is then rotated clockwise by 6 positions, so that the vertex t... | 2 | MATH |
[
{
"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 uncolored, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is uncolored, vertex 7 is red, vertex 8 is red. How many valid colorings of the uncolored... | 0 | MATH |
[
{
"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 red, 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 vertices that can be ma... | 3 | MATH |
[
{
"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 uncolored, vertex 3 is blue, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist ... | 5 | MATH |
[
{
"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 blue, vertex 6 is red, vertex 7 is blue. What is the maximum number of red vertices that can be made to occupy a po... | 3 | MATH |
[
{
"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 blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is blue. How many valid colorings of the uncolored vertices exist s... | 1 | MATH |
[
{
"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 uncolored, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is blue, vertex 7 is blue. How many valid colorings of the uncolored vertices exist ... | 2 | MATH |
[
{
"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 blue, 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 red vertices that can be mad... | 3 | MATH |
[
{
"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 uncolored vertices exist su... | 7 | MATH |
[
{
"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 red, vertex 6 is red, vertex 7 is blue. What is the maximum number of red vertices that can be made to occupy a ... | 2 | MATH |
[
{
"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 blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red, vertex 8 is red. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"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 red, vertex 5 is red, vertex 6 is red, vertex 7 is blue. What is the maximum number of red vertices that can be made to occupy a p... | 3 | MATH |
[
{
"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 blue, vertex 6 is red, vertex 7 is red. The 8-gon is then rotated clockwise by 2 positions, so that the vertex that... | 2 | MATH |
[
{
"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 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 made to occupy a pos... | 3 | MATH |
[
{
"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 red. What is the maximum number of blue vertices that can be made to occupy a ... | 3 | MATH |
[
{
"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 red, vertex 6 is blue, vertex 7 is red, vertex 8 is red. What is the maximum number of red vertices that can be ma... | 3 | MATH |
[
{
"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 red, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is red, vertex 7 is uncolored, vertex 8 is blue. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"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 uncolored, vertex 5 is blue, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colorings of the uncolo... | 0 | MATH |
[
{
"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 blue, vertex 6 is blue, vertex 7 is red, vertex 8 is red. The 9-gon is then rotated clockwise by 5 positions, so ... | 2 | MATH |
[
{
"content": "A 9-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 red, vertex 4 is blue, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is uncolored, vertex 8 is blue. How many valid colorings of the uncolor... | 4 | MATH |
[
{
"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 blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue. What is the maximum number of blue vertices that can be... | 3 | MATH |
[
{
"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 blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is uncolored. How many valid colorings of the uncolor... | 1 | MATH |
[
{
"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 uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is uncolored, vertex 7 is uncolored, vertex 8 is blue. How many valid colorings of the uncolore... | 2 | MATH |
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