Norman Do The Art of Tiling with Rectangles 1

Tiling pervades the art and architecture of various ancient civilizations. Toddlers grapple with tiling problems when they pack away their wooden blocks and home renovators encounter similar problems in the bathroom. However, rather than being a frivolous pastime, mathematicians have found the art of tiling to be brimming with beautiful mathematics, problems of fiendish difficulty, as well as important applications to the physical sciences. In this article, we will consider some of the more surprising results from the art of tiling with rectangles. One of the most famous of tiling conundrums is the following, a problem which almost every mathematician must have encountered at one time or another. Consider the regular 8 × 8 checkerboard which has been mutilated by removing two squares from opposite corners. How many ways are there to tile the remaining board with dominoes which can cover two adjacent squares? The answer to this problem, which may seem surprising to an unsuspecting audience, is that it is impossible to tile the mutilated checkerboard. Prior to removing the two squares, there is a myriad of ways to perform such a domino tiling — actually, 3604 = 12988816 ways to be precise! So why should such a trivial alteration of the board reduce this number to zero? The argument is stunning in its simplicity and the key to the solution lies in the seemingly unimportant colouring of the checkerboard into black and white squares. Of course, this colouring is such that the placement of any domino on the board will cover exactly one square of each colour. Thus, a necessary condition for the board to be tiled by dominoes is that there are an equal number of black and white squares. However, in mutilating our checkerboard, we have removed two squares of the same colour from a board that previously had 32 of each. From this disparity, we are led to the conclusion that the mutilated checkerboard cannot be tiled by dominoes, no matter how hard one might try.