# Puzzles

## 21 December

To get today's number, multiply the number of modules needed to make a dodecahedron by half the number of tube maps used to make a stellated rhombicuboctahedron.

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One module is needed for each edge of the dodecahedron. Therefore 30 modules are needed.

48 tube maps are needed for the stellated rhombicuboctahedron.

Therefore today's number is 720.

## Folding tube maps

Back in 2012, I posted instructions for

folding a tetrahedron from tube maps. When tube maps are used, the sides of the tetrahedron are not quite equal. What ratio would the rectangular maps need to be in to give a regular tetrahedron?

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Once the map is folded, it will look like this:

For the final tetrahedron to be regular, the red lengths must be equal. Let each red length be 2 (this will get rid of halves in the upcoming calculations). By drawing a vertical line in we can work out the width and height of the rectangle:

The width of the rectangle is 3 (one and a half red lengths). Using Pythagoras' Theorem in the blue triangle, we find that the height of the rectangle is \(\sqrt{3}\). Therefore, the ratio of the rectangle is \(\sqrt{3}:3\) or \(1:\sqrt{3}\).

#### Extension

If the ratio of the rectangle is \(1:a\), what is the ratio of the lengths of the sides of the tetrahedron?