Problem:
A rectangular floor that is feet wide and feet long is tiled with one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and last tile, how many tiles does the bug visit?
Answer Choices:
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Solution:
Because and are relatively prime, the diagonal does not cross the boundaries between tiles at any corner point of the tiles. In order for the bug to move from one vertex of the rectangle to the opposite vertex, the bug must cross edges in one direction and edges in the other direction, a total of edges. Each time the bug crosses an edge, it enters a new tile. Counting the tile it started on as well, the bug visits a total of tiles.
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The problems on this page are the property of the MAA's American Mathematics Competitions