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.


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.


What if applet does not run?

Discussion

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Copyright © 1996-2017 Alexander Bogomolny

The 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:

two L-trominoes tile a 2x3 rectangle

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 ab. 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 ab, 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 ≤ ab.

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, and 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 ≤ ab. An a×b rectangle can be tiled with L-trominoes if and only if one of the following conditions holds:

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

Reference

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

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  • Golomb's inductive proof of a tromino theorem
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  • Covering a Chessboard with a Hole with L-Trominoes
  • Tromino Puzzle: Deficient Squares
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  • Tiling a Square with T-, L-, and a Square Tetrominoes
  • Tiling a Rectangle with L-tetrominoes
  • Tiling a 12x12 Square with Straight Trominoes
  • Bicubal Domino
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