# The Inside-Out Puzzle

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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)