3360.22 – City Hall Tile Floor
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?
Take a piece of graph paper with 1" squares, or a checkerboard. Stretch a thread across a "floor" of various small dimensions--2 x 2, 2 x 3, 2 x 4, 5 x 2, etc, and tally the number of squares that are crossed. It makes a nice class activity for pairs of students, one stretching the thread, both counting, and the other writing.
What emerges is a fascinating and non-obvious pattern. For a floor that is n x m tiles, the number of tiles the mouse crosses is equal to m + n minus the greatest common factor of m and n.
For the floor that's 93 x 231, the answer is 93 + 231 - 3 = 321.
N.B. The problem assumes that the tiles are lined up with the walls, so that the floor looks like some sort of a checkerboard. This is not the proper way to lay the tiles (ask the Bostonians): they should be laid on the diagonal to look right.