The Gold Coin game

Let us play the Gold Coin game―with proper play, you can become rich! We shall make some strategic insights, but eventually we shall realize the game is really Nim in disguise.

Let us play the Gold Coin game, a two-player moving-coin game played with a gold coin and some pennies on a linear board. Each player strives to get the gold coin, worth far more than all the pennies combined. Can you try to win the gold coin?

The game begins with the coins on the track. The gold coin is farthest to the right, and during play all the coins will move only to the left. On each turn, a player may either (1) move any coin to the left by one or more spaces but without jumping onto or over another coin, or (2) take the leftmost coin into his or her possession. Whoever takes the gold coin is the winner―the pennies just don’t matter. In the diagram above, one of the players has just moved the gold coin three spaces. We shall see in the end that this was a very good move, a winning move, and indeed it is the only winning move available in this position.

What do you think is the winning strategy?

Interlude

I would recommend playing the game with a partner several times in order to gain familiarity with the rules and how game play proceeds. Here is one way the game indicated above, for example, might have proceeded in a sequence of legal moves.

The resulting position is this:

What is the best move to play next?

Perhaps at this stage it may not be clear exactly how to play, for the game may seem to have an open-ended nature, with moves taking place in an enormous game space of possible play, too complex for easy analysis. In the end, however, we shall on the contrary learn how to conceive of the game in a way that makes the winning moves completely transparent. We will be able to look at any position and easily find the winning moves.

In this last position, for example, knowing this winning perspective I can tell you easily that there are exactly two correct winning moves, and all others will be losing. One winning move will be to move the leading penny up two squares, like this:

And the other winning move will be to move the middle penny up two squares, like this:

All other moves, we will see, are losing moves, leading inevitably to a loss against any skilled opponent. By the end of the chapter we shall explain why these two moves and only these two moves are winning in this position.

Please enjoy this selection from Infinite Games: Frivolities of the Gods, my new book serialized here with fresh material on games and the logic of games each week.

This week we look into the Gold Coin game.