By Natalie Hartley
Do you know how maths can be used to invent new juggling tricks or why a single game called Nim is the best and simplest way to study game theory?
On the fifth of December, Year 12 Mathematics students visited the Maths in Action event, where these questions were answered. Six talks were presented to us, including, ‘How to Get Rich Quick’, ‘Risk and Reward’ and ‘Magical Maths’. The aim of these talks was to give A-Level Mathematics students an opportunity to learn where Maths could be taken beyond schooling. This provided a useful insight on six completely different pathways where Maths had played a huge role in the reason they were able to speak to us.
The first talk was by Colin Wright, who explained that most classic two-player games can be reduced to a game called Nim to allow for mathematical analysis. Nim is a game that involves a player-selected number of piles and sums in each pile. The game is turn-based and the aim of the game is to remove any number of the sum from one of the piles until it is impossible for the other player to remove any more of the sum, causing them to lose. Nim with its simple form allows the optimum player strategy to be examined.
To examine the strategy, Colin Wright simplified the number of each pile into binary. For example, the number 13, is shown in the table below.
| Binary | 8 | 4 | 2 | 1 |
| 14 | 1 | 1 | 0 | 1 |
| 9 | 1 | 0 | 0 | 1 |
| 5 | 0 | 1 | 0 | 1 |
| 11 | 1 | 0 | 1 | 1 |
| Total | 3 | 2 | 1 | 4 |
Next, he labelled position ‘P’ as the position where the previous player would win and position ‘N’ as the position where the player about to play would win. He determined that for player ‘P’ to cause player ‘N’ to lose, the total of each of the columns would have to be even numbers.
Due to the game being simple, it allows game theory to be studied on a deeper level due to it being easier to track both players playing the ideal moves.
More information can be found on this JStor page, where the original paper about game theory was written: https://www.jstor.org/stable/1967631?origin=crossref
Lastly, Colin Wright explained that he charted all the juggling tricks he could do using a table. He said that each ball had three states: before being thrown in the air, being thrown in the air and after it had been caught. Tracking all of the juggling tricks he could do in this manner meant that the combinations he didn’t already have tricks for were new tricks waiting to be discovered. He performed a short juggling show, displaying these tricks.