A classroom contains 25 desks arranged in 5 rows and 5 columns. The teacher asks each student to move to the desk in front of, behind, to the left of, or to the right of her current desk. The students at the edges have limited choices — will every child be able to find a new seat?
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No, they won’t. Think of the classroom as a 5 × 5 checkerboard. Each student must move to a square of a new color, but there are unequal numbers of light and dark squares, so the task is impossible. If there were an even number of rows or columns, it could be done.
From Ross Honsberger, In Pólya’s Footsteps, 1997.
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