# Tiling Rectangles with L-Trominoes

L-tromino is a shape consisting of three equal squares joined at the edges to form a shape resembling the capital letter L. There is a fundamental result of covering an N×N square with L-trominos and a single monomino. Here we are concerned with a different question: what kind of rectangles can be completely tiled with L-trominos?

The applet below helps you experiment with the problem. It takes three clicks to place an L-tromino on the board. Click on yet uncovered square and move (not drag) the cursor to an availble neighbor. Click and move the cursor to complete an L-shape at another available square.

What if applet does not run? |

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Copyright © 1996-2018 Alexander BogomolnyThe following two observations must be obvious:

First of all, since an L-tromino consists of 3 squares, a rectangle tiled by a number of L-trominoes must consist of a multiple of 3 squares. So, for example, a 4×5 rectangle could not be tiled by L-trominoes. This means that, for a rectangle to be tiled with L-trominoes, one of its sides must be divisible by 3. This is a *necessary* but not a *sufficient* condition. Indeed, rectangles 1×3, 3×3 and 5×3 could not be so tiled, as can be easily seen with the help of the applet.

Secondly, the 2×3 is the smallest rectangle that could be tiled by L-trominoes:

Alexander Soifer splits the proof of the main result concerning the general case of the tiling with L-trominoes, into a sequence of exercises:

**Exercise 2.2** Find the smallest square that can be tiled with L-trominoes.

**Exercise 2.3** Find all integers *b* such that a 2×*b* rectangle can be tiled using L-trominoes.

**Exercise 2.4** Find all integers *b* such that a 3×*b* rectangle can be tiled using L-trominoes.

Combining the two last exercises we obtain the following

### Theorem 2.1

Let *a*, *b* be the integers such that 2 ≤ *a* ≤ 3 and *a* ≤ *b*. An *a*×*b* rectangle can be tiled by L-trominoes if and only if *ab* is divisible by 6.

(This is equivalent to saying that, with 2 ≤ *a* ≤ 3 and *a* ≤ *b*, one of the numbers *a*, *b* is divisible by 2 and the other by 3.)

The next sequence of exercises applies to the case where 4 ≤ *a* ≤ *b*.

**Exercise 2.5** Prove that a 5×6 rectangle can be tiled using L-trominoes.

**Exercise 2.6** Prove that a 5×9 rectangle can be tiled using L-trominoes.

**Exercise 2.7** Prove that a 9×9 rectangle can be tiled using L-trominoes.

**Exercise 2.8** Prove that that if *b* > 5 and *b* is divisible by 3 then a 5×*b* rectangle can be tiled using L-trominoes.

**Exercise 2.9** Prove that that if *b* is divisible by 3 and an *a*×*b* rectangle can be tiled L-trominoes, then this is also true for an *a* + 2)×*b*.

**Exercise 2.10** Let the integers *a*, *b* satisfy the conditions *a* ≥ 4,*b* ≥ 5,*b* is divisible by 3. Decide whether an *a*×*b* rectangle can be tiled L-trominoes.

### Theorem 2.2

An *a*×*b* rectangle where *a* ≥ 4, *b* ≥ 4 can be tiled with L-trominoes if and only if *ab* is divisible by 3.

Combining the statements of Theorems 2.1 and 2.2 gives

### Theorem 2.3

Let a, *b* be the integers such that 2 ≤ *a* ≤ *b*. An *a*×*b* rectangle can be tiled with L-trominoes if and only if one of the following conditions holds:

- a = 3 and b is even;
- a ≠ 3 and ab is divisible by 3.

### Reference

- S. Golomb,
*Polyominoes*, Princeton University Press; 2nd edition (March 18, 1996) - A. Soifer,
*Geometric Etudes in Combinatorial Mathematics*, Springer, 2010 (2nd, expanded edition), pp 7-9

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