Tangram – the puzzle pieces

The Tangram is an old Chinese dissection puzzle and probably the best-known puzzle featured on this site. It consists of seven pieces of different shapes and sizes.

The basic unit of each Tangram piece is a right isosceles triangle. Shapes made from these triangles are called polyaboloes (also known as polytans). The two small triangles each consist of one unit triangle; the medium triangle, square, and parallelogram each consist of two unit triangles; and the two large triangles each consist of four unit triangles.

The Tangram Puzzle

Where can you get a Tangram puzzle?

Tangram puzzles made of wood or cardboard are available in stores. You can also make one easily with a pair of scissors and a sheet of paper.

This page includes cardboard templates for making a puzzle and 3D-printing templates to download.

Tangram figure made from cardboard pieces

Cardboard and 3D-printed Tangram sets arranged on a figure template — Cardboard and 3D-printed Tangram sets

Grid alignment

Thousands of different figures can be made with the Tangram. Some outlines, such as the square, fit exactly into a grid of right isosceles triangles.

For a grid-aligned solution, the grid must be fine enough: the small Tangram triangle must consist of four grid triangles. In addition, the edges of all the pieces must run along grid lines. In some solutions, the outline fits the grid, but at least one piece edge does not follow a grid line; those solutions are not grid-aligned.

Tangram figure in the shape of a cat — This cat figure is not grid-aligned on the grid shown.

Non-grid-aligned solution of the Tangram square — Non-grid-aligned solution

Grid-aligned solution of the Tangram square — Grid-aligned solution

Solver program

A solver program for Tangram puzzles is available to download for these grid-aligned figures.

The program grid is fine enough to represent every solution to a given figure.

A selection of Tangram figures

Convex Tangram Figures

Definition:
A geometric shape is convex if the line segment between any two points in the shape lies entirely inside it.

There are only 13 convex Tangram figures. Fu Traing Wang and Chuan-Chih Hsiung proved this in 1942. On the refined grid, all 13 convex figures are grid-aligned.