South African Mathematics Olympiad Third Round

Question

Zola and Ron play a game by alternately moving a single ten cent coin on a circular board. The game starts with the ten cent coin already on the board as shown. A player may move the coin either clockwise one position or one position toward the centre, but cannot move to a position that has been previously occupied.

The last person who is able to move wins the game.

If Zola starts, which player can play in a way that guarantees a win?

Explain this player's winning strategy.

Accepted Answer

Since the game must end (finite number of blocks and can't play on a previously occupied block) and only one person can move last, there will be a winner and hence a winning strategy.

We claim that the first player (Zola) has the winning strategy and it is to always move clockwise. There are

7

positions left in the outer ring so only player

1

(Zola) can complete the outer ring and then player

2

(Ron) will be forced to go inwards. Note at any stage player

2

(Ron) can opt to go inwards, but if he moves inwards, there will be

7

positions left in that ring and again only player

1

(Zola) can complete that ring. This is the same for each ring. Hence only player

1

(Zola) can complete the inner ring and move last and hence win.