B3. Twelve balls are numbered with the numbers $1,2,3, \ldots, 12$. Each ball is painted either red or green such that the following conditions are met:

(a) if the balls labeled with different numbers $a$ and $b$ are painted red and $a+b<13$, then the ball labeled with the number $a+b$ is also painted red;

(b) if the balls labeled with different numbers $a$ and $b$ are painted green and $a+b<13$, then the ball labeled with the number $a+b$ is also painted green.

In how many ways can the balls be painted?

## Problems for 3rd Grade

Time for solving: 120 minutes. In section A, a correct answer will be worth two points, while a wrong answer will result in a deduction of half a point. Write your answers for section A in the left table.

![](https://cdn.mathpix.com/cropped/2024_06_07_4050f14f84cf4f327f67g-06.jpg?height=160&width=317&top_left_y=548&top_left_x=641)

| B1 | B2 | B3 |
| :--- | :--- | :--- |
|  |  |  |
B3. Suppose ball 1 is red. If ball 2 is also red, then red are also $1+2=3, 1+3=4, \ldots, 1+11=12$. Therefore, all balls are red.

If ball 2 is green, we again consider two cases. If ball 3 is red, then red are also $1+3=4, 1+4=5, \ldots, 1+11=12$, so all other balls are red. If ball 3 is green, then $2+3=5$ is also green. Since $1+4=5$ and ball 5 is green, and ball 1 is red, ball 4 cannot be red, so it must be green. Then, balls with numbers $4+2=6, 5+2=7, 6+2=8, \ldots, 9+2=11$, $10+2=12$ are also green, so all other balls are green.

We have three possibilities. If we swap the roles of green and red, we get three more colorings. In total, there are 6 ways: all balls are red, all balls are green, all balls except ball 1 are green, all balls except ball 1 are red, all balls except ball 2 are red, and all balls except ball 2 are green.

There are 6 correct colorings, 4 points (If the contestant does not list all solutions, award 1 point for the first correct coloring found, 2 points for 2 or 3 colorings, 3 points for 4 or 5 colorings.) Explanation of cases or justification showing that there are no other correct colorings besides the six listed 2 points

## Solutions for 3rd grade

In section A, a correct answer will be awarded 2 points, while an incorrect answer will result in a deduction of 0.5 points. To avoid a potential negative final score, each contestant is given an initial 2 points.

| A1 | A2 | A3 |
| :---: | :---: | :---: |
| E | E | C |

## Justifications: