My wife and I like to play backgammon. Recently we sat in a pub in the UK and wanted to play, but didnâ€™t have the equipment. We figured that it shouldnâ€™t be impossible to improvise a decent backgammon game with materials that are readily available to an average person sitting in a pub. The materials we ended up using were: paper, a pen, the pub menu, and coins. We drew the backgammon board on a sheet of paper, using the pub menu as a substitute for a ruler. As playing pieces we used small pieces of paper with an x symbol for one player and an o symbol for the other. Figure 1 shows the actual backgammon board we used.

We decided to use coins as substitute for the dice, but ran into a logical puzzleâ€”what system based on coins to use that will provide an adequate alternative to dice? We wanted a system that:

1. Has six different possible states that would represent for us the numbers 1 through 6.
2. In each round, produces a random state with the same probability for each state to be chosen.

Can you think of a system that meets the puzzleâ€™s requirements? It doesnâ€™t have to incorporate exactly three coins as is seen in Figure 1, but it does have to meet both requirements presented above. BTW, if you can think of a system that would substitute dice without coins, rather with other readily available items to an average person in a pub, Iâ€™d be most interested to hear your idea.

Cheers,
BG