A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is [i]chesslike[/i] if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board.

[i]Proposed by Nikola Velov[/i]
To solve this problem, we need to find the maximal number of $2 \times 2$ chesslike subsquares on a $6 \times 6$ board where no four adjacent unit squares are of the same color in any horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare is chesslike if it has one red and one green diagonal.

1. **Understanding the Constraints:**
   - The board is $6 \times 6$, so it has $36$ unit squares.
   - No four adjacent unit squares can be of the same color in any direction.
   - A $2 \times 2$ subsquare is chesslike if it has one red and one green diagonal.

2. **Counting the Total Number of $2 \times 2$ Subquares:**
   - A $6 \times 6$ board has $(6-1) \times (6-1) = 5 \times 5 = 25$ possible $2 \times 2$ subsquares.

3. **Maximizing Chesslike Subquares:**
   - We need to ensure that as many of these $25$ subsquares as possible are chesslike.
   - A chesslike $2 \times 2$ subsquare has the following pattern:
     \[
     \begin{array}{|c|c|}
     \hline
     R & G \\
     \hline
     G & R \\
     \hline
     \end{array}
     \quad \text{or} \quad
     \begin{array}{|c|c|}
     \hline
     G & R \\
     \hline
     R & G \\
     \hline
     \end{array}
     \]

4. **Constructing the Board:**
   - We can use a checkerboard pattern to ensure that no four adjacent squares are of the same color.
   - For example, we can color the board as follows:
     \[
     \begin{array}{|c|c|c|c|c|c|}
     \hline
     R & G & R & G & R & G \\
     \hline
     G & R & G & R & G & R \\
     \hline
     R & G & R & G & R & G \\
     \hline
     G & R & G & R & G & R \\
     \hline
     R & G & R & G & R & G \\
     \hline
     G & R & G & R & G & R \\
     \hline
     \end{array}
     \]

5. **Counting Chesslike Subquares in the Checkerboard Pattern:**
   - In this pattern, every $2 \times 2$ subsquare is chesslike.
   - Therefore, all $25$ possible $2 \times 2$ subsquares are chesslike.

6. **Verifying the Constraints:**
   - The checkerboard pattern ensures that no four adjacent squares are of the same color in any direction.
   - Thus, the pattern satisfies all the given constraints.

7. **Conclusion:**
   - The maximal number of chesslike $2 \times 2$ subsquares on the $6 \times 6$ board is $25$.

The final answer is $\boxed{25}$.