My wife and I like to play backgammon. Recently we sat in a pub in the UK and wanted to play, but didn’t have the equipment. We figured that it shouldn’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 puzzle—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 puzzle’s requirements? It doesn’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, I’d be most interested to hear your idea.

Cheers,
BG