Tetromino — Definition, Formula & Examples
A tetromino is a geometric shape formed by joining exactly four equal-sized squares edge to edge. There are five distinct tetrominoes, famously recognized as the falling pieces in the game Tetris.
A tetromino is a polyomino of order 4 — a connected plane figure composed of four unit squares joined along their edges. Up to rotation and reflection (free tetrominoes), there are exactly 5 distinct tetrominoes: the I, O, T, S, and L shapes.
How It Works
To find all tetrominoes, start with a single square and systematically add squares one at a time, always sharing a full edge. Two shapes that can be rotated or flipped to match each other count as the same free tetromino. This process yields exactly 5 free tetrominoes. If you treat rotations as different (one-sided tetrominoes), you get 7. If you also treat reflections as different (fixed tetrominoes), you get 19.
Example
Problem: Can you tile a standard 4 × 5 rectangle using all 5 free tetrominoes exactly once?
Count the squares: Each tetromino covers exactly 4 unit squares, so 5 tetrominoes cover a total of:
Check the rectangle: A 4 × 5 rectangle contains exactly 20 unit squares, so the areas match. A tiling is at least possible in principle.
Verify a solution exists: By arranging the I, O, T, S, and L tetrominoes (allowing rotation and reflection), you can indeed tile the 4 × 5 rectangle with no gaps or overlaps. Multiple valid arrangements exist.
Answer: Yes. Since the total area matches (20 squares) and valid arrangements can be found, the 4 × 5 rectangle can be tiled using all 5 free tetrominoes exactly once.
Visualization
Why It Matters
Tetrominoes appear in competition math problems involving tiling, covering, and combinatorial enumeration. They also introduce key ideas in discrete geometry, such as counting distinct shapes under symmetry transformations — a gateway to the mathematical field of group theory.
Common Mistakes
Mistake: Counting rotations and reflections as separate tetrominoes and getting more than 5.
Correction: When counting free tetrominoes, two shapes that match after any rotation or flip are the same shape. This gives exactly 5 distinct tetrominoes, not 7 or 19.