Tangram – Pentoma Puzzles

Tangram – The Game Pieces

The ancient Chinese dissection puzzle Tangram is probably the best-known of the games presented here. It consists of seven puzzle pieces of different sizes and shapes.

The basic shape of the pieces is a right isosceles triangle. The puzzle pieces therefore belong to the family of polytans (polyabolos). The two small triangles consist of one triangle each; the medium triangle, the square, and the parallelogram consist of two triangles each; and the two large triangles consist of four triangles each.

The Tangram as a Puzzle

Where can you get a Tangram puzzle?

The Tangram puzzle is available commercially in wooden and cardboard versions. However, it is also easy to make your own. All you need is a pair of scissors and a sheet of paper.

This page offers crafting templates for a cardboard puzzle and 3D-printer templates available for download.

Grid conformity

There are thousands of different shapes for the Tangram. A few of them – such as the square – fit exactly into a grid made of right isosceles triangles.

For a shape to fit completely into the grid, the grid must be chosen fine enough: each small Tangram triangle must consist of four grid triangles. Otherwise, many solutions are not grid-conform – meaning the overall shape fits into the grid, but the edges of individual pieces do not follow the grid lines.

tangram shape cat — this cat does not fit into any grid

non–grid-conforming solution of the tangram square — non–grid-conforming solution

grid-conforming solution of a tangram square — grid-conforming solution

Solver program

For such grid-conforming shapes, this page provides a Tangram puzzle solver available for download.

The program’s grid is fine enough to represent all solutions of any given shape within the grid.

A selection of Tangram shapes

Convex shapes of the Tangram

Definition:
A geometric shape is convex if, for any two points A and B within the shape, the entire line segment connecting A and B also lies within the shape.

For the Tangram puzzle, there are only 13 convex shapes (proved by Fu Traing Wang and Chuan-Chih Hsiung in 1942). All convex shapes are grid-conforming when using the refined grid.