You ordered a pizza for your party, but the restaurant forgot to slice it – these mathematical tricks can help you cut it evenly, says Katie Steckles.

Fairness is important – in life, and in pizza. If you want to cut a pizza into equal-sized pieces, the difficulty will depend on how many people you need to share it between. Luckily, mathematics has some tricks to keep things equal.

For example, if the number of people you are sharing a pizza between is a power of two – one, two, four, eight, 16 – cutting the pizza into as many slices is easy. For one piece, obviously no cuts are needed. For each larger power of two, a cut across through the centre of the pizza – cutting all of the existing pieces exactly in half – will result in pieces of equal size.

Some numbers will be much harder: prime numbers, by definition, can’t be divided easily. Luckily, geometry can help.

If you need to cut a pizza into five equal pieces, first grab a long, thin, rectangular strip of paper. Tie the paper in an ordinary overhand knot, like you would tie in a piece of string. Then, keeping the ends flat, pull gently to tighten the knot. The whole thing will flatten and come together – stop pulling when you can’t go any further without it wrinkling.

The flat shape you are looking at should now be vaguely familiar, if you ignore the two ends of paper sticking out. Fold these ends into the middle, or cut them off, and you will have a shape with five straight edges, created purely by the shape of the knot. Yes, that is right – you have made a perfect regular pentagon, with five equal-length sides and five equal angles at the corners.

It is possible to prove this mathematically by showing that all the folds you make in the paper strip are at 72 degrees to the parallel edges of the strip. But for simplicity, because the paper is the same width everywhere, and weaves in and out five times in the right way, these will be five equal edges. And more importantly, the pentagon’s corners are equally spread around a circle – making it the perfect guide for pizza slicing.

Place your pentagon in the centre of the pizza, then cut along lines radiating out from the centre of the pentagon and through each corner. And presto: you have a pentagonal pizza party for five. This paper-strip method can be used whenever you are in a pentagon-based emergency.

You can use the same technique to produce a shape with any odd number of sides by creating a more complex knot with the strip passing through the middle more times, although the strip of paper needs to be increasingly thin and it takes a lot more patience to pull the ends through and carefully flatten out the shape.

Combined with our existing halving methods, you can now produce any number of slices you like. The same results can be extended to any other round food – thanks to maths, the world is your cheesecake.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to **Katie Steckles ***