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Tromino as a Rep-tileThe applet below illustrates a proof of S. Golomb's theorem based on the fact that the L-shaped (or right tromino) is a rep-4 tile. (The term rep-tile has also been introduced by Golomb to describe a shape that is tiled by smaller copies of itself.)
TheoremA unit square has been removed from a 2n×2n board. The rest of the board can be tiled with L-shaped trominos. By clicking anywhere in the applet you can define (and redefine) the missing square. The "exponent" parameter defines n, for a 2n×2n board. The number can be changed by clicking a little off it central line.
Copyright © 1996-2009 Alexander Bogomolny
ProofDivide the board into 4 equal squares. The missing square belongs to one of the four. The other three form an L-shaped tromino, which, being a rep-4 tile, can be tiled by smaller copies of itself, which, in turn, can be tiled (if necessary) by smaller copies of themselves, and so on. Every size board can be treated in this manner. In particular, we can apply this step to the remaining square where the small missing square is located. The process will stop when References
Copyright © 1996-2009 Alexander Bogomolny |
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