In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones?

[b]Note[/b]. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells.
1. **Identify the constraints**: We are given a $12 \times 12$ square table with stones placed such that each row, column, and diagonal has an even number of stones. Each diagonal is parallel to one of the two main diagonals of the table and consists of $1, 2, \ldots, 11$ or $12$ cells.

2. **Analyze the diagonals**: Note that the diagonals of the table can be categorized into two types:
   - Diagonals parallel to the main diagonal (from top-left to bottom-right).
   - Diagonals parallel to the anti-diagonal (from top-right to bottom-left).

3. **Count the diagonals**: There are $24$ odd-length diagonals in total. This is because each main diagonal has $12$ diagonals, and among these, the lengths alternate between odd and even. Specifically, the lengths of the diagonals are $1, 3, 5, 7, 9, 11$ (6 odd-length diagonals) and $2, 4, 6, 8, 10, 12$ (6 even-length diagonals) for each main diagonal direction. Thus, there are $6 + 6 = 12$ odd-length diagonals in each direction, totaling $24$ odd-length diagonals.

4. **Determine the placement of stones**: Each odd-length diagonal must have at least one "hole" (absence of stone) to ensure an even number of stones. Since there are $24$ odd-length diagonals, there must be at least $24$ holes.

5. **Calculate the maximum number of stones**: The total number of cells in the $12 \times 12$ table is $144$. Given that there must be at least $24$ holes, the maximum number of stones is:
   \[
   144 - 24 = 120
   \]

6. **Verify the configuration**: To check if this bound is tight, consider a configuration where every cell not on a "main diagonal" is filled. This means we place stones in all cells except those on the main diagonals. This configuration ensures that each row, column, and diagonal has an even number of stones, satisfying the problem's conditions.

Conclusion:
The maximum number of stones that can be placed on the $12 \times 12$ table, while ensuring that each row, column, and diagonal has an even number of stones, is $\boxed{120}$.