In short. A polyomino is a shape made of unit squares joined edge to edge. Four squares give the tetrominoes — the seven Tetris pieces. Five give the twelve pentominoes, the classic puzzle set. Fitting a set of them into a fixed outline is the entire genre of block-fitting games, and it is a hard problem: the general case is NP-complete.
How many are there?
The count grows fast, and no formula is known for it — the numbers below come from exhaustive computer enumeration, counting shapes as identical when one is a rotation or reflection of another.
| Squares | Name | How many |
|---|---|---|
| 1 | monomino | 1 |
| 2 | domino | 1 |
| 3 | tromino | 2 |
| 4 | tetromino | 5 |
| 5 | pentomino | 12 |
| 6 | hexomino | 35 |
| 7 | heptomino | 108 |
| 8 | octomino | 369 |
The five tetrominoes become the familiar seven of Tetris because that game treats mirror images as different pieces — S and Z, J and L are reflections that cannot be rotated into each other on a flat screen.
The twelve pentominoes
The pentominoes are the set the genre is built on, and they carry letter names for their shapes: F, I, L, N, P, T, U, V, W, X, Y, Z. Together they cover exactly 60 squares, which is why the classic challenge is to tile a 6×10, 5×12, 4×15 or 3×20 rectangle with all twelve.
The 6×10 rectangle has 2 339 distinct solutions. The 3×20 has just 2. That collapse — same pieces, same area, a thousandfold difference in how forgiving the frame is — is the clearest illustration of what makes one board of a fitting game trivial and another cruel.
Why fitting is hard
Deciding whether a set of polyominoes tiles a given region is NP-complete. There is no clever rule that reads a board and tells you the answer; past a certain size, the honest method is to try, backtrack, and try again.
When rotation is not allowed, the problem changes character rather than getting simpler. Each piece has exactly one orientation, so the search is smaller — but you lose the usual escape of turning an awkward piece to make it fit, and a hole of the right area may have no solution at all.
The parity trick
One tool does let you rule out impossible puzzles without searching. Colour the target region like a chessboard, then count.
Some pieces always cover the same number of dark and light squares whatever their position; others always cover an unequal split. Add up what your set must cover and compare it with what the region actually has. If the totals cannot match, no arrangement exists — and you have proved it in thirty seconds instead of exhausting the search.
It is the same argument that shows a chessboard with two opposite corners removed cannot be covered by dominoes: the two missing squares share a colour, so 30 of one and 32 of the other can never be paired.
Reading a fitting puzzle
- Place the awkward pieces first. Long straights and crosses have the fewest positions that work; the small compact ones fit almost anywhere and can wait.
- Start at the corners and edges. They constrain the most and give you certainty early.
- Watch for the single orphan square. A gap of one square that nothing can reach means the arrangement is already dead — restart rather than continue.
Frequently asked questions
What is a polyomino?
A shape formed by joining unit squares edge to edge. A domino is two squares, a tetromino four — the Tetris pieces — and a pentomino five. There are twelve distinct pentominoes.
How many pentominoes are there?
Twelve, counting rotations and reflections of the same shape as one piece. They are named after the letters they resemble: F, I, L, N, P, T, U, V, W, X, Y and Z, and together they cover exactly 60 squares.
Why can I not fit the pieces even though the area is right?
Matching area is necessary but not sufficient. A colouring argument often proves it outright: shade the region like a chessboard and count what each piece must cover — if the totals cannot balance, no arrangement exists at all.
Is fitting polyominoes into a shape a hard problem?
Yes, formally. Deciding whether a set of polyominoes tiles a given region is NP-complete, so there is no general shortcut — beyond small boards the reliable method is systematic trial and backtracking.
What changes when pieces cannot be rotated?
Each piece has a single orientation, which makes the search space smaller but the puzzle less forgiving: you cannot turn an awkward shape to rescue a placement, so a hole of the correct area may simply have no solution.
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