## Maths Is Magic

By Alycia Winter, University of South Australia

I have always been passionate about mathematics. This is due to how you can find it anywhere. You use it when you shop or when you cook. It is also hidden in less obvious things like music or a shell of a snail. One of my favourite uses for maths is magic.

Magic is an ancient past time, which has been mesmerising people around the world for centuries. Magic comes in all shapes and forms, all of which are based on illusion and sleight of hand, meant to deceive an audience into believing the impossible. Behind all these tricks there are simple explanations, which allow the magician to captivate others. Mathematics is the basis behind a range of tricks, both the simple and the more involved.

One type of trick that relies of maths is card tricks. The Perfect Shuffle is commonly utilized by magicians to perform a range of magic tricks which relies on the mathematical concept of arrangement to have its unique cyclic property. The Reed-Solomon code magic trick which relies on field theory. Card algebra is a small trick that relies on basic algebra to “guess” an audience members card.

Another card trick, or more so an idea which is utilized in tricks is the Gilbreath Principle which is a card theorem. This card trick uses a deck of cards arranged in alternating colours. The deck is then cut in two so that the bottom card of each deck is a different colour. These halves are then riffle shuffled together. Pairs of cards are taken from the top of this deck and it is seen that every pair consists of a black and red card.

This trick can be shown to work with inductive proof. Assume that the first card to fall, in the riffle shuffle, is a black card. If the next card to fall is from the same half then it will be red. Similarly if it is from the other deck, the card will also be red. Thus there will be a red black pair. This leaves the cards in the same initial condition, with opposite colours sitting on the base of each half. This allows the same logic to able to be applied to each dropping of cards.

A similar result can be gained with the four suits in a deck, as the Gilbreath Principle generalises. It could even be extended to a group of five cards or even a double deck using a similar method to the original.

Alycia Winter was one of the recipients of a 2016/17 AMSI Vacation Research Scholarship.