Tromino as a Rep-tile

The 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.)

Theorem

A 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 with a spin control at the bottom of the applet.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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Proof

A unit square has been removed from a 2ⁿ×2ⁿ board. The rest of the board can be tiled with L-shaped trominos.

Proof

Divide 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 n = 1, i.e., when a 2×2 square is reached that conisists of a single L-tromino and the missing square.

References

S. Golomb, Two Right Tromino Theorems, in The Changing Shape of Geometry, edited by C. Pritchard, Cambridge University Press, 2003
S. Golomb, Polyominoes, Princeton University Press, 1994

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Tromino as a Rep-tile

Theorem

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Proof

References

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Related material
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