Tromino tiling by divide and conquer

An L-tromino covers three squares in an L shape. A classic result: any $2^n \times 2^n$ board with a single square removed can be tiled by L-trominoes. The proof is divide and conquer. Split the board into four quarters, place one tromino at the shared centre so it covers one square in each of the three quarters that lack the hole, and now every quarter is a smaller board with one square removed. Repeat.

Counting gives a quick sanity check: a tromino covers $3$ squares, so the number of squares to tile must be a multiple of $3$. For $2^n \times 2^n$ minus one square that count is $4^n - 1$, which is always divisible by $3$.