# janmr blog

## Tiling with L-Trominos January 24, 2016

An L-tromino is a figure in the plane made of three equal-sized squares connected in an L-shape:

Consider now the following question: Given a $2^n \times 2^n$ square grid ($n \geq 0$) with exactly one cell occupied, is it possible to tile the remaining area using L-trominos?

The answer is yes and it is quite easy to see why. First, if it is a $1 \times 1$ grid ($n=0$) with one cell occupied then there is nothing to tile and we are done.

For $n \geq 1$ we can divide the square into four sub-squares each of size $2^{n-1} \times 2^{n-1}$. The occupied cell must appear in exactly one of these sub-squares and we now place an L-tromino appropriately at the center such that one cell gets occupied in each of the other three sub-squares:

Now each of the sub-squares have exactly one cell occupied and the theorem follows by induction.

As an example, here is a complete tiling for a $32 \times 32$ grid:

The theorem is mentioned in Proof Without Words II by Roger B. Nelsen and in Mathematical Gems III by Ross Honsberger. The theorem itself is attributed to Solomon W. Golomb.