Ltromino

Ltromino is a polyomino consisting of three unit squares arranged to form an L shape: two adjacent squares along one axis with a third square attached at one end perpendicularly. In tiling theory it is commonly referred to as the L-tromino or L-triomino. The shape is a member of the triomino family and has a 2-by-2 bounding box with one cell missing.

In the plane, the Ltromino has four rotational orientations. In the free-polyomino sense, reflections do not yield a distinct shape, since the L-tromino is congruent to its mirror image by a rotation.

Tilings and puzzles: The L-tromino is famous for the tromino tiling problem. A 2^n by 2^n board with one square removed can be tiled by L-trominoes for every n ≥ 1. The standard proof uses induction: place a central L-tromino to cover the three central squares surrounding the missing square in the board's central area, leaving four smaller quadrants, each with exactly one square missing, and tile recursively.

Area considerations: Each Ltromino covers three unit squares, so any region tiled with Ltrominoes must have area divisible by 3. Not all regions with area multiple of 3 admit a tiling; shape constraints apply.

History and references: The L-tromino is one of the classic polyomino shapes studied in tiling theory. It appears in several combinatorics texts; a foundational treatment is found in Golomb's Polyominoes (1965).