The Inside-Out Puzzle

Introduction

This intriguing little puzzle was first shown to me by my high school maths teacher. It consists of a ring of four squares of card. Each square is divided into four by its diagonals. The inside and outside of the ring are given different colours, and the object is to turn the ring inside-out so that the colours are reversed. Flexing is only permitted along the diagonals and the joins between the squares; the 16 small triangles should remain planar.

Discussion

The problem is harder than you might think. I have heard that there are three ways to perform the reversal. I have definitely found two distinct ways. One of these has two equivalent enantiomeric (mirror image) variants. I do not know if these should be counted separately to give the total of three, or if there is another way I have yet to find. However, I have certainly done better than Cundy and Rollett [1], who only managed to find one way.

The solutions that I have found are symmetrical in time; a set of moves is used to get to a half way point, and then the sames moves are reversed to get back to the ring. The half-way stages have some symmetry wih respect to the two colours; loosly the 'inside' and 'outside' are as much black as they are white.

Contruction Advice

I am not aware of this puzzle being on sale anywhere, but it is fairly straightforward to make your own. To make it easier to play with, you need to make the flexing lines very flexible, and the rest of the model as rigid as possible.

My model is made from (about 1mm) thick card which has ben cut into the 16 small triangles. The squares sides are about 7cm in length. The triangles are fixed back together with sticky tape, leaving a gap of about 3mm between the card pieces. The latter is vital, to allow the folding of one part around another.

References

[1] Cundy H.M. & Rollett A.P., Mathematical Models (third edition), §4.9.2. Tarquin Publications, 1981. (ISBN 0906212200)