Problem 16 · 2025 Math Kangaroo
Hard
Logic & Word Problems
caseworksymmetry
We consider a giant \(4 \times 4\) chessboard. A kangaroo is standing on each of the 16 squares. On each move, each kangaroo jumps to an adjacent square (up, down, left or right, but not diagonally). All kangaroos stay on the chessboard. Several kangaroos can be on one square at the same time. What is the maximum number of unoccupied squares that we can have after 100 moves?
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Answer: B — 14
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Hint 1 of 2
Colour the board like a checkerboard; what happens to a kangaroo's colour each jump?
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Hint 2 of 2
Every jump flips colour, so after an even number of moves the 8 dark-start and 8 light-start kangaroos stay split—each group can pile onto one square.
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Approach: checkerboard parity invariant
- Each jump changes a kangaroo's square colour, so after 100 (even) moves 8 kangaroos sit on dark squares and 8 on light squares.
- Each group can be gathered onto a single square, occupying just 2 squares total.
- Maximum unoccupied = 16 − 2 = 14.
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