Dissection puzzles: from the tangram to polyominoes

Last updated: 15 July 2026

A dissection puzzle is the oldest trick in recreational geometry: cut a shape into pieces, and ask someone to make something of them. People have been playing this game for at least two centuries. Undivide belongs to that lineage — with one small twist that gives it its name.

The tangram craze

The most famous dissection puzzle is the tangram: a square cut into seven pieces — five triangles, a square, and a parallelogram. It emerged in China around the turn of the 19th century and, within a couple of decades, swept Europe and America as a genuine craze; publishers sold books of hundreds of silhouettes to reproduce, and everyone from schoolchildren to (as the legend goes) Napoleon in exile spent evenings sliding the seven pieces around.

The tangram's formula is the classic one: a fixed set of pieces, an endless catalogue of target silhouettes. The pieces never change; the goal always does.

Puzzle columns and a haberdasher

A century later, dissection became a playground for the great newspaper puzzlists — Sam Loyd in America and Henry Dudeney in England. Dudeney's haberdasher's problem (1902) is still the genre's showpiece: an equilateral triangle cut into just four pieces that reassemble into a perfect square. Hinge the pieces at their corners and the triangle swings open into the square in one motion.

Mathematics eventually explained why such tricks are always possible: the Wallace–Bolyai–Gerwien theorem (proved in the early 19th century) says that any polygon can be cut into finitely many pieces and rearranged into any other polygon of the same area. Every dissection puzzle is a small, hands-on demonstration of that theorem.

Polyominoes: dissection goes onto a grid

In 1953, the mathematician Solomon Golomb gave a talk that named a new family of shapes: polyominoes — figures made of unit squares glued edge to edge, the way two squares make a domino. Martin Gardner popularized them in his Scientific American column, and the five-square pentominoes became a classic in their own right: twelve pieces, and the long-standing challenge of tiling rectangles with them. (Their best-known descendant, of course, is the four-square family that fell from the top of the screen in 1984.)

Putting dissection on a square grid changed its character. Pieces snap rather than slide; a fit is exact or it is nothing; and questions like "how many ways can these pieces tile this shape?" become precise enough for mathematics — and for software — to answer exactly.

Undivide turns the form inside out

The classic formula is fixed pieces, endless targets. Undivide inverts it: the figure comes first, and the cut itself is the puzzle-maker. Each level takes a solid figure and deterministically dissects it into connected polyomino pieces, scrambles them into a tray — rotated, flipped, shuffled — and asks you to undo the division. Hence the name: you un-divide.

Because the cut is generated, no two levels need ever repeat. Every level is produced from a seed — a number that fully determines the figure, the cut, and the scramble — so the supply is endless, the difficulty can ramp gently and precisely, and the same seed always reproduces the same level. Any valid tiling of the figure counts as solved; you are not hunting for the one "intended" arrangement, just a way to make the figure whole.

And since the pieces live on a grid with exactly eight possible orientations each, restoring the figure is really a chain of small mental rotations — which is the quiet skill the whole game is built around. The rest of the experience — no timers, no fail states, a knowable minimum number of moves for every level — is covered in A calm puzzle game, by design.

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