Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.

\[\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
\hline
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\
\hline
25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 \\
\hline
33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\
\hline
41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 \\
\hline
49 & 50 & 51 & 52 & 53 & 54 & 55 & 56 \\
\hline
57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 \\
\hline
\end{tabular}\]

Assume we have an infinite supply of knights. We place knights in the chess board squares such that no two knights attack one another and compute the sum of the numbers of the cells on which the knights are placed. What is the maximum sum that we can attain?

Note. For any $2\times3$ or $3\times2$ rectangle that has the knight in its corner square, the knight can attack the square in the opposite corner.
1. **Understanding the Knight's Movement**:
   - A knight in chess moves in an "L" shape: two squares in one direction and then one square perpendicular, or one square in one direction and then two squares perpendicular.
   - This means a knight placed on a square can attack up to 8 different squares, depending on its position on the board.

2. **Coloring the Chessboard**:
   - We can color the chessboard in a checkerboard pattern with alternating black and white squares.
   - This ensures that a knight placed on a black square can only attack white squares and vice versa.

3. **Placing Knights on the Chessboard**:
   - To maximize the sum of the numbers on the squares where knights are placed, we need to place knights on either all black squares or all white squares.
   - This is because knights on the same color squares do not attack each other.

4. **Calculating the Sum for Black and White Squares**:
   - The chessboard is an 8x8 grid, so there are 32 black squares and 32 white squares.
   - The numbers on the chessboard range from 1 to 64.

5. **Sum of Numbers on Black Squares**:
   - The black squares are those where the sum of the row and column indices is even.
   - We can list the numbers on the black squares and sum them up:
     \[
     \begin{aligned}
     &1, 3, 5, 7, 10, 12, 14, 16, 17, 19, 21, 23, 26, 28, 30, 32, \\
     &33, 35, 37, 39, 42, 44, 46, 48, 49, 51, 53, 55, 58, 60, 62, 64
     \end{aligned}
     \]
   - The sum of these numbers is:
     \[
     \sum_{\text{black squares}} = 1 + 3 + 5 + 7 + 10 + 12 + 14 + 16 + 17 + 19 + 21 + 23 + 26 + 28 + 30 + 32 + 33 + 35 + 37 + 39 + 42 + 44 + 46 + 48 + 49 + 51 + 53 + 55 + 58 + 60 + 62 + 64 = 1056
     \]

6. **Sum of Numbers on White Squares**:
   - The white squares are those where the sum of the row and column indices is odd.
   - We can list the numbers on the white squares and sum them up:
     \[
     \begin{aligned}
     &2, 4, 6, 8, 9, 11, 13, 15, 18, 20, 22, 24, 25, 27, 29, 31, \\
     &34, 36, 38, 40, 41, 43, 45, 47, 50, 52, 54, 56, 57, 59, 61, 63
     \end{aligned}
     \]
   - The sum of these numbers is:
     \[
     \sum_{\text{white squares}} = 2 + 4 + 6 + 8 + 9 + 11 + 13 + 15 + 18 + 20 + 22 + 24 + 25 + 27 + 29 + 31 + 34 + 36 + 38 + 40 + 41 + 43 + 45 + 47 + 50 + 52 + 54 + 56 + 57 + 59 + 61 + 63 = 1056
     \]

7. **Conclusion**:
   - Since the sum of the numbers on the black squares is equal to the sum of the numbers on the white squares, the maximum sum we can attain by placing knights on either all black squares or all white squares is 1056.

The final answer is \(\boxed{1056}\).