Polyominoes

How many are there?

The word itself comes from the mathematician Solomon Golomb, who named the family in 1953 and turned it into a research topic in its own right.

The count grows fast and no formula is known for it; the numbers below come from exhaustive computer enumeration. Two shapes count as one when a rotation or a reflection turns one into the other.

Tetris uses seven pieces rather than five because it treats mirror images as distinct: S and Z, and J and L, are reflections, and on a flat screen no rotation turns one into the other.

The parity trick

One check rules out impossible puzzles before you place a single piece. Colour the target region like a chessboard, then count.

Some pieces cover the same number of dark and light squares wherever they sit; others cover an unequal split, which can tip one way or the other depending on where they sit. Run through what your set of pieces can cover and compare it with what the region actually has. If the totals cannot match, no arrangement exists.

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, leaving 30 of one and 32 of the other, and every domino takes one of each.

Reading a fitting puzzle

• Place the awkward pieces first. Long straights and crosses often have the fewest positions that work; the small compact ones fit almost anywhere and can wait.
• Start at the corners and edges. They have the fewest ways to be covered, so they settle early.
• Watch for a stranded single square. A one-square gap that nothing can fill means the arrangement is already dead, so restart instead of continuing.

The twelve pentominoes

The twelve pentominoes carry letter names for their shapes: F, I, L, N, P, T, U, V, W, X, Y, Z. Together they cover 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, counting those that map onto each other by symmetry only once. The 3×20 has 2. Same pieces, same area: the shape of the frame alone accounts for the difference, and it is what makes one board of a fitting game forgiving and another almost rigid.

Why fitting is hard

Deciding whether a set of polyominoes tiles a given region is NP-complete. Past small boards there is no known shortcut: a classic approach is to place a piece, backtrack when the placement fails, and try the next one.

Woodfit, as the rules above say, never rotates or mirrors a piece. That shrinks the search, since every piece has a single orientation, but it also removes the usual fix of turning an awkward shape, so a hole of the right area can still have no solution.

In this game

Controls

Drag a piece from the tray into the carved hollow. A ghost shows where it lands when the spot is legal; otherwise the blocking cells turn red and the piece goes back to the tray. Tap a placed piece to pop it back, or drag it straight elsewhere. Top bar: back on the left, hint bulb and restart on the right, they fire when you lift your finger, so slide off to cancel.

Rules

Pieces are never rotated or mirrored: they go in exactly as they sit in the tray. Every hollow has a solution. Three stars means no take-back at all. The hint bulb frees the fewest blocking pieces and places one for good, a rewarded video in the app, and it costs you no star. Progress is kept server-side and follows you to a new phone.

Frequently asked questions