The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.

Find the number of possible colorings that satisfies the above conditions.
1. **Initial Assumptions and Definitions:**
   - We are given that there is a bijection between black and white numbers.
   - The sum of three black numbers is black.
   - The sum of three white numbers is white.

2. **Case Analysis:**
   - We start by assuming \(1\) is black. By induction, we will show that all odd numbers must be black.
   - Suppose \(1\) is black. Then, \(1 + 1 + k = k + 2\) must be black for all odd \(k\). This implies that all odd numbers are black.

3. **Considering Even Numbers:**
   - Suppose there exists a black even number. By the well-ordering principle, let \(m\) be the smallest black even number.
   - By induction, \(1 + 1 + k = k + 2\) must be black for all even \(k \ge m\). This implies that all even numbers greater than or equal to \(m\) are black.
   - Since \(m\) is the smallest black even number, all even numbers less than \(m\) must be white.

4. **Contradiction with Bijection:**
   - There are infinitely many odd numbers, hence infinitely many black numbers.
   - If there are only finitely many white numbers (as there are only finitely many even numbers less than \(m\)), a bijection between black and white numbers cannot exist.
   - Therefore, the assumption that there exists a black even number is false. Hence, all even numbers must be white.

5. **Alternative Coloring:**
   - If we assume \(1\) is white instead, by similar reasoning, all odd numbers must be white, and all even numbers must be black.

6. **Conclusion:**
   - There are exactly two possible colorings that satisfy the given conditions:
     1. All odd numbers are black, and all even numbers are white.
     2. All odd numbers are white, and all even numbers are black.

The final answer is \(\boxed{2}\)