City Hall Mouse, Numerical

City Hall has a splendid rectangular black-and-white tiled floor. The tiles are squares. There are 93 of them in one direction and 231 in the other. A mouse runs in a straight line diagonally from one corner of the floor to the opposite corner. How many tiles does it cross?

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This problem may be best tackled by starting with much smaller grids, drawing diagonals across the grids, counting the number of tiles touched by each diagonal, and looking for a pattern. You should eventually arrive at the formula:

For a floor that is m by n tiles, the diagonal (or the running mouse) touches m + n – (the GCF of m and n) tiles.

Thus, for the floor at City Hall, the mouse will cross 93 + 231 – 3 = 321 tiles.

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