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# Practice Paper 3 Question 18

Let $$A$$, $$B$$ and $$C$$ represent the number of tokens in three piles. A $$\textit{game}$$ between Alice and Bob starts with at most $$n$$ tokens in each pile (i.e. $$0 < A, B, C \leq n$$) and consists of them taking turns. A $$\textit{turn}$$ consists of a player removing $$x$$ tokens from one pile and $$y$$ tokens from a different pile, with the constraint that $$x + y > 0.$$ The player to remove the last set of tokens wins. Alice goes first in every game, and they play through all possible starting configurations of $$A, B, C.$$ If Alice and Bob play in such a way that ensures their own number of wins is maximized, how many games does Alice win?

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## Warm-up Questions

1. Alice and Bob play tic-tac-toe. Bob starts with an $$X$$ in the top left corner, and Alice replies with an $$O$$ in the opposite corner. How should Bob play to win?
2. $$31$$ tokens lie on the table. In each move, a player can take one or two coins. The player to remove the last token wins. Alice decided to leave her opponent with a number of tokens which is divisible by $$3$$ after each move. Prove, using mathematical induction, that she will be able to do that, no matter what her opponent does, and finally win the game.

## Hints

• Hint 1
Who wins when $$A=B=C=1$$ and why?
• Hint 2
Describe all configurations from which you can get to $$A=B=C=1$$ in one move.
• Hint 3
Notice that in all of them you can ensure a win.
• Hint 4
If $$A, B, C$$ are not all equal, can you make one move to make them all equal?
• Hint 5
Who wins when $$A=B=C=2$$ and why?
• Hint 6
Generalise your result to any case when $$A=B=C$$?
• Hint 7
Use the principle of mathematical induction to prove that all positions when $$A=B=C$$ are losing and that all others are winning.

## Solution

For $$A=B=C=1$$ case Alice loses as she can remove either 1 or 2 tokens, leaving Bob to remove the remaining ones. Notice that whoever has $$A=B=C$$ loses and otherwise can win. We may use the following observations to build an inductive proof of this fact.

• $$(1, 1, 1)$$ is a losing position.
• When $$A, B, C$$ are not all equal, one can choose the smallest and make the remaining ones equal to it.
• When $$A, B, C$$ are all equal, it is impossible to keep them all equal, due to the constraint $$x + y > 0.$$

For all games starting with $$A=B=C$$, Bob wins, and for all other games, Alice wins. There are $$n^3$$ games in total, excluding 0 games, and Alice wins $$n^3-n$$ games, and Bob wins $$n$$.

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