What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers?
1. **Assumptions and Definitions**:
   - We are given an \( n \times n \) chessboard.
   - A checker occupies one square on the chessboard.
   - A checker may only be placed on a square, and a square may only have at most one checker.
   - We need to find the maximum number of checkers such that each row and each column contains an even number of checkers.

2. **Case 1: \( n \) is even**:
   - If \( n \) is even, we can place a checker in every square of the chessboard.
   - This results in \( n^2 \) checkers.
   - Since \( n \) is even, each row and each column will contain \( n \) checkers, which is an even number.
   - Therefore, the maximum number of checkers is \( n^2 \).

3. **Case 2: \( n \) is odd**:
   - If \( n \) is odd, we can place a checker in every square except for one main diagonal.
   - This results in \( n^2 - n \) checkers.
   - Each row and each column will contain \( n-1 \) checkers, which is an even number.
   - To prove that \( n^2 - n + 1 \) checkers cannot be placed:
     - By the Pigeonhole Principle, distributing \( n^2 - n + 1 \) checkers to \( n \) rows means at least one row will have \( n \) checkers.
     - Since \( n \) is odd, having \( n \) checkers in a row results in an odd number of checkers in that row, which contradicts the requirement.
   - Therefore, the maximum number of checkers is \( n^2 - n \).

4. **General Formula**:
   - For even \( n \), the maximum number of checkers is \( n^2 \).
   - For odd \( n \), the maximum number of checkers is \( n^2 - n \).

The final answer is \( \boxed{ 2n \cdot \left\lfloor \dfrac{n}{2} \right\rfloor } \). In other words, if \( n \) is even, the answer is \( n^2 \), and if \( n \) is odd, the answer is \( n^2 - n \).