Primes of the L tetromino

[L tetromino]

2 × 4 (smallest rectangle)
3 × 8

smallest rectangle: 2 × 4

[2 x 4 rectangle]

Proposition. The L tetromino is even.
Proof. Number the squares by

                (x, y) |---> { 1  if  x  is even
                             { 0  if  x  is odd

No matter how it is placed, an L tetromino covers an odd total. However, any rectangle with area divisible by 4 covers an even total. Therefore, if it is tiled by L tetrominoes, there must be an even number of them. QED.

Golomb [1] asks for all rectangles which can be tiled by the L tetromino. This proposition is the key to Klarner's solution. He gives a more general criterion using the same numbering in [2, Theorem 4].


[1a] S.W. Golomb, Covering a Rectangle with L-tetrominoes, Problem E 1543, American Mathematical Monthly 69 (1962) p. 920.
[1b] Solution to Problem E 1543 by D.A. Klarner, American Mathematical Monthly 70 (1963) pp. 760-761.
[2] David A. Klarner, Packing a Rectangle with Congruent N-ominoes, Journal of Combinatorial Theory 7 (1969) pp. 107-115.

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Updated August 23, 2011.