Solution
L-Trominoes in a Strip
Show the problem again
An L-tromino is the shape made of three unit squares forming an L (a 2 × 2 square with one cell removed). In how many ways can a 2 × 12 rectangle be tiled exactly by L-trominoes, each of which may be rotated or reflected freely? Tilings that differ by a symmetry of the rectangle count as different. Give an integer.
Worked solution
**Answer: 16.** In a strip of height 2, every L-tromino occupies two adjacent columns: both cells of one column and one cell of the other. Look at the leftmost column. Its two cells cannot belong to two different trominoes, because each would then need both cells of column 2. So a single tromino covers column 1 and one cell of column 2. The other cell of column 2 must then be covered by a tromino that also takes both cells of column 3. Those two pieces fill a 2 × 3 block exactly. Hence every tiling is a sequence of 2 × 3 blocks, and each block can be filled in exactly 2 ways (the two pieces form mirror-image arrangements). With 12 / 3 = 4 blocks, there are 2^4 = 16 tilings.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.