## 18 Apr 2013

### The Sylver Coinage Game: Upper Secondary Mathematics

This is more a project than a single question. There is, however, a question at the end; and it will help you with tomorrow’s PMQ.

John Conway invented a number of mathematical games including one that has come to be called the Sylver Coinage game. I usually try to obfuscate the standard names of various theorems so that users actually try to solve the problems rather than searching for the answer. In this case, the game remains interesting even after reading the literature.

The game is played by two players; let’s call them Alice and Bill. The game starts with the set of all positive whole numbers. Alice picks the first number – this removes from the game all multiples of this number. Bill then picks a number – this removes from the game all linear combinations of the two numbers in play. The game continues in this fashion until one player is left having to pick the 1 – that player is the loser.

Before we experiment with a couple of games, let me go into the way we shall write each move. We shall use curly brackets to show the sequence of moves, so that {5, 7} means that Alice picks the 5 followed by Bill picking the 7. You may also come across square brackets; these indicate known winning moves. For example, the game {5, 7} has [8]. We shall see why in a moment. To add some clarity to the available options, I am going to use standard brackets to show all possible legal moves remaining. So, as we seem to have already started a game, let’s play!

Alice picks 5. Bill picks 7.
{5, 7} (1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18, 23) 12 numbers left
Alice picks 8.
{5, 7, 8} (1, 2, 3, 4, 6, 9, 11) 7 numbers left
What can Bill do now? He cannot pick the 2 or 3, otherwise Alice will then pick whichever of 2 or 3 is left and win. Bill picks 11.
{5, 7, 8, 11} (1, 2, 3, 4, 6, 9)
Alice picks 9.
{5, 7, 8, 11, 9} (1, 2, 3, 4, 6)
It should be obvious that Bill will lose this. Whoever is the first one to pick either the 2 or 3 will lose. If Bill picks the 4 or 6, Alice picks whichever one of those two is left and Bill is forced to make a losing move. Let’s fast-forward to the end. Bill picks 3.
{5, 7, 8, 11, 9, 3} (1, 2, 4)
Alice picks 2.
{5, 7, 8, 11, 9, 3, 2} (1)
Bill loses! Alice wins!

It is a good idea to replay this game so that you gain some experience and confidence in how the game works. Why is the 8 a winning second move for Alice? What happens if Alice doesn't pick 8 on her second move?

Mercifully, you’ll be pleased to know that there is an application that simplifies the calculations. The "Sylver Coinage" CDF can be found at the Wolfram Demonstrations Project. You will need to download Wolfram’s CDF Player and then can either play it within your browser or download the CDF and play it offline.

The Question

We finally come to a question! Assume that both Alice and Bill are rational players and both trying to win. Alice moves first and picks 7; Bill then picks 9. The game after two moves is written as {7, 9}.

Find a winning strategy for Alice to win, whatever moves Bill makes.

Rather than giving all possible games, you can submit just one game but showing that Bill’s moves are the best possible.

There is a theorem that states that the first player will always win - with one condition. What do you think that condition is?