Pentomino — Definition, Formula & Examples
A pentomino is a flat shape formed by connecting exactly 5 equal-sized squares, where each square shares at least one full edge with another square in the group. There are 12 unique pentomino shapes, and each is named after a letter of the alphabet it resembles (F, I, L, N, P, T, U, V, W, X, Y, Z).
A pentomino is a polyomino of order 5 — a connected, edge-joined arrangement of five unit squares in the plane. Two pentominoes are considered the same if one can be rotated or reflected to match the other. Under this equivalence, there are exactly 12 distinct free pentominoes.
How It Works
To build a pentomino, start with one square and keep attaching squares edge-to-edge until you have five. Two arrangements count as the same pentomino if you can flip or rotate one to match the other. Since each of the 12 pentominoes covers exactly 5 square units, all 12 together cover square units. A classic puzzle challenge is to tile a rectangle using all 12 pentominoes with no gaps or overlaps.
Example
Problem: Can all 12 pentominoes fit perfectly into a 6 × 10 rectangle with no gaps or overlaps?
Step 1: Count the total area covered by all 12 pentominoes.
Step 2: Find the area of the rectangle.
Step 3: Since the areas match, a tiling is possible in principle — and in fact, there are exactly 2,339 distinct solutions to this puzzle.
Answer: Yes. The areas match at 60 square units, and 2,339 valid tilings exist.
Why It Matters
Pentomino puzzles build spatial reasoning skills used in geometry and design. The board game Blokus is based directly on polyominoes including pentominoes, making them one of the most widely played examples of recreational mathematics.
Common Mistakes
Mistake: Counting rotations or reflections as separate pentominoes, leading to more than 12 shapes.
Correction: A pentomino that can be flipped or rotated to match another is considered the same piece. With this rule, there are exactly 12 unique pentominoes.