1998 USAMO Problems/Problem 4

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A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.


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Answer: 98.

There are 4·97 adjacent pairs of squares in the border and each pair has one black and one white square. Each move can fix at most 4 pairs, so we need at least 97 moves. However, we start with two corners one color and two another, so at least one rectangle must include a corner square. But such a rectangle can only fix two pairs, so at least 98 moves are needed.

It is easy to see that 98 suffice: take 49 1x98 rectangles (alternate rows), and 49 98x1 rectangles (alternate columns).

credit: https://mks.mff.cuni.cz/kalva/usa/usoln/usol984.html

editor: Brian Joseph

See Also

1998 USAMO (ProblemsResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6
All USAMO Problems and Solutions

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