S. Golomb gave an inductive proof to the following fact: any 2n×2n board with one square removed can be tiled by trominos - a piece formed by three adjacent squares in the shape of an L. The applet below helps you test your understanding of the theorem by tiling the board manually. It takes three clicks to place a tromino piece on the board: click on three adjacent squares in sequence. (Do not drag the mouse.)
|What if applet does not run?|
Note that for small (2×2 and 4×4) boards the solution is unique and follows Golomb's proof. For larger boards, viz. starting with 8×8, this is no longer true: there is a good deal of solutions. However, some thinking is still required to tile the board and, more often than not, careless tiling will produce 1 and 2 squares pockets.
Another applet provides additional insight into the tiling with L-trominoes.
Interestingly, we run into an entirely different situation if we try to cover the chessboard with straight trominos. Now, we'll have to consider very carefully which single square may or may not be removed!