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A $$7 \times 7$$ board is divided into 49 unit squares. We place checkers on the board, at most one per square. Find the largest number of checkers that can be placed on the unit squares so that each row, as well as each column, contains an even number of checkers.
Correct Answer: 42
A row has 7 squares, so the largest even number of checkers it can contain is 6, and with seven rows the total cannot exceed $$7 \times 6 = 42$$. This bound is attained by leaving the seven main-diagonal squares empty and putting a checker in every other square, since each row and each column then holds exactly 6 checkers. Hence the answer is 42.
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