Hexaflexagons are paper polygons with a surprising number of faces.

They were first discovered in 1939 by Arthur Stone, who set up a Flexagon Committee to investigate their properties.

The committee consisted of the famous Mathematicians and Physicists, B. Tuckerman, R.P. Feynman and J.W. Tukey.

Trihexaflexagons

The trihexaflexagon has three different faces. To make a trihexaflexagon follow these easy steps.

 Step 1: Cut a strip of paper and mark ten equilateral triangles as shown on the left. Number the triangles and crease along the lines joining the triangles. Fold the strip back along the line marked ab. Step 2: Fold the strip back along the line marked cd. Step 3: Fold the triangle labelled 1 to the back and glue it to the back of the triangle labelled 3*. Step 4: To change the face of your trihexaflexagon, place your thumb and forefinger on the two triangles labelled 2* and pinch them together so their backs are touching. From the centre ease out the inner edge of the two unmarked triangles so that your trihexaflexagon is flat.

Now you can have some fun. Colour the three different faces of your trihexaflexagon with different designs and start flexing. You should find that there are a few more designs than you counted on!

Hexahexaflexagons

A hexahexaflexagon has six different faces.

Construct a hexahexaflexagon as follows:

 Step 1: Take a longer piece of paper (almost twice as long as the trihexaflexagon) and mark 19 equilateral triangles as shown. Carefully label the triangles on both sides of the paper (as shown above) and crease along the lines joining the triangles. Step 2: Fold so that the triangles labelled 4 face each other, as do the triangles labelled 5 and the triangles labelled 6. You are essentially wrapping the paper into a spiral. Your strip should now resemble that used to make the trihexaflexagon. Step 3: Fold and glue as in the instructions for the trihexaflexagon. You should find that all the triangles on each side have the same number.

Start flexing and see if you can find all the other sides. (Some sides are trickier to find than others.) Colour or paste pictures on the six different faces and flex away. How many different patterns can you make?