Flexagons are a favorite mathematical craft. The original flexagon—the trihexaflexagon—appears to be a two-sided hexagon, but by performing a pinch flex move, the hexagon opens from the center displaying a hidden face. Continuing to pinch flex the shape and we see that the flexagon has three sides.

That is not quite true, however. While it is true that if we were to color the sides of the trihexaflexagon, we would need three colors, each colored side can appear on the front or on the back of the flexagon. And when the face moves from front to back, the arrangement of the rhombus-shaped pieces change orientation relative to each other. So in a sense, the trihexaflexagon has six faces.

I took advantage of this property to make the following Halloween trihexaflexagon. You can download a printable pdf.

Here are the six faces.

Here is a video of the flexagon in action.

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A post shared by Dave Richeson (@divisbyzero)

Instructions for making the FLEX-A-GHOUL:

• Print the template and cut out the flexagon pattern.
• Color the template (better now so that the ink—if you use markers—doesn’t bleed through to the other side).
• Fold in half lengthwise and glue or tape back-to-back.
• Although the pattern shows rhombuses, really you should think about it as a string of 10 equilateral triangles. Fold and unfold along the edges of each triangle so they are creased.
• Fold the “TRICK” rhombus in half along its diagonal, and do the same with the “TREAT” and “OR” rhombuses.
• Finally, the two end triangles are both blank on one side. Glue or tape these blank sides together.