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?
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.