Box $A$ contains black balls and box $B$ contains white balls. Take a certain number of balls from $A$ and place them in $B$. Then take the same number of balls from $B$ and place them in $A$. Is the number of white balls in $A$ then greater, equal to, or less than the number of black balls in $B$?
1. **Define Variables:**
   Let \( X \) be the number of black balls in box \( A \) and \( Y \) be the number of white balls in box \( B \). Let \( q \) be the number of balls transferred from box \( A \) to box \( B \).

2. **Transfer Balls from \( A \) to \( B \):**
   When \( q \) black balls are taken from box \( A \) and placed in box \( B \), the number of black balls in box \( A \) becomes \( X - q \), and the number of black balls in box \( B \) becomes \( q \).

3. **Transfer Balls from \( B \) to \( A \):**
   Now, we take \( q \) balls from box \( B \) and place them in box \( A \). Let \( p \) be the number of black balls and \( b \) be the number of white balls taken from box \( B \), such that \( p + b = q \).

4. **Analyze the Transfer:**
   - If \( p = q \), then all \( q \) balls taken from box \( B \) are black. Thus, the number of black balls in box \( A \) becomes \( X - q + q = X \), and the number of black balls in box \( B \) becomes \( 0 \).
   - If \( 0 < p < q \), then \( 0 < q - p = b \). This means some of the balls taken from box \( B \) are white. The number of black balls in box \( A \) becomes \( X - q + p \), and the number of black balls in box \( B \) becomes \( q - p \).
   - If \( p = 0 \), then all \( q \) balls taken from box \( B \) are white. Thus, the number of black balls in box \( A \) becomes \( X - q \), and the number of black balls in box \( B \) remains \( q \).

5. **Compare the Number of White Balls in \( A \) and Black Balls in \( B \):**
   - After the transfer, the number of white balls in box \( A \) is \( b \), which is the number of white balls taken from box \( B \).
   - The number of black balls in box \( B \) is \( q - p \), which is the number of black balls remaining in box \( B \).

6. **Conclusion:**
   Since \( b = q - p \), the number of white balls in box \( A \) is equal to the number of black balls in box \( B \).

\[
\boxed{\text{The number of white balls in } A \text{ is equal to the number of black balls in } B.}
\]