Problem 1312

1. Show that the negative of a white number must be colored black:

Suppose $a$ is white. According to the given rules:
- The sum of two white numbers must be black. Therefore, $a + a = 2a$ is black.
- The negative of a black number must be white. Therefore, $-2a$ is white.
- Now consider $a - 2a = -a$. Since $a$ is white and $-2a$ is white, their sum $-a$ must be black.

Hence, if $a$ is white, then $-a$ must be black.

2. Show that the sum of two black numbers must be colored white:

Suppose $a$ and $b$ are black. According to the given rules:
- The negative of a black number must be white. Therefore, $-a$ and $-b$ are white.
- The sum of two white numbers must be black. Therefore, $-a + (-b) = -(a + b)$ is black.
- The negative of a black number must be white. Therefore, $a + b$ must be white.

Hence, the sum of two black numbers must be white.

3. Determine all possible colorings of the integers that satisfy these rules:

Let's analyze the possible colorings:
- From the rules, we know that if $a$ is white, then $-a$ is black.
- If $a$ is black, then $-a$ is white.
- The sum of two white numbers is black.
- The sum of two black numbers is white.

Consider the number $0$:
- If $0$ is white, then $0 + 0 = 0$ must be black, which is a contradiction.
- If $0$ is black, then $0 + 0 = 0$ must be white, which is also a contradiction.

Therefore, $0$ cannot be colored either black or white. This implies that $0$ must be the third color, red.

Now, consider the implications:
- If $a$ is any non-zero integer, it must be either black or white.
- If $a$ is white, then $-a$ is black.
- If $a$ is black, then $-a$ is white.

This leads to the conclusion that all non-zero integers must be either black or white, and $0$ must be red.

Therefore, the only possible coloring that satisfies all the given rules is:
- $0$ is red.
- All positive integers are either black or white.
- All negative integers are colored such that the negative of a black number is white and vice versa.

The final answer is $\boxed{ 0 }$ is red, and all non-zero integers are either black or white, with the negative of a black number being white and vice versa.