Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?
1. **Understanding the Problem:**
   We need to place checkers on an $8 \times 8$ chessboard such that each row and each column contains twice as many white checkers as black ones. 

2. **Constraints:**
   - Each row and each column can have at most 6 checkers.
   - The ratio of white to black checkers in each row and column must be 2:1.

3. **Mathematical Formulation:**
   Let $w_i$ and $b_i$ be the number of white and black checkers in the $i$-th row, respectively. Similarly, let $w_j$ and $b_j$ be the number of white and black checkers in the $j$-th column, respectively. We need:
   \[
   w_i = 2b_i \quad \text{and} \quad w_j = 2b_j \quad \text{for all } i, j.
   \]

4. **Maximum Checkers in a Row/Column:**
   Since each row and column can have at most 6 checkers, we can write:
   \[
   w_i + b_i \leq 6 \quad \text{and} \quad w_j + b_j \leq 6.
   \]
   Given $w_i = 2b_i$, we substitute to get:
   \[
   2b_i + b_i \leq 6 \implies 3b_i \leq 6 \implies b_i \leq 2.
   \]
   Therefore, the maximum number of black checkers in any row or column is 2, and the maximum number of white checkers is:
   \[
   w_i = 2b_i = 2 \times 2 = 4.
   \]

5. **Total Checkers:**
   Each row and each column can have at most 2 black checkers and 4 white checkers. Since there are 8 rows and 8 columns, the total number of black checkers is:
   \[
   8 \times 2 = 16.
   \]
   The total number of white checkers is:
   \[
   8 \times 4 = 32.
   \]

6. **Example Configuration:**
   The given example configuration is:
   - Column 1: Row 4 (B), Row 6 (B)
   - Column 2: Row 1 (B), Row 5 (B)
   - Column 3: Row 2 (B), Row 6 (B)
   - Column 4: Row 1 (B), Row 5 (B)
   - Column 5: Row 2 (B), Row 6 (B)
   - Column 6: Row 3 (B), Row 7 (B)
   - Column 7: Row 4 (B), Row 8 (B)
   - Column 8: Row 3 (B), Row 7 (B)

   This configuration ensures that each row and each column has exactly 2 black checkers and 4 white checkers.

Conclusion:
The maximum number of checkers that can be placed on the board, satisfying the given conditions, is:
\[
\boxed{48}
\]