[pdurrant]

This is an adaption of a classic puzzle, in two parts and with three questions.

You're in the Bazaar of Honest Conmen. Yes, really — here there are numerous stall where you can play the shell game in various forms, but the showmen don't cheat. It's actually possible to find the lady, pick the right shell, or, as in the stall we're going to look at, find the coin. There's no sleight of hand or trickery, the games work just as they appear to do on the surface.

You wander through looking for a game to play. Obviously, you're only going to be risking money you'd intended to spend on entertainment, but you're still keen to win if you can.

You come up to a stall, where on the table top are three identical, upside-down, tall, hollow cylinders. You watch for a while, and you see that the game is to find a pound coin hidden under one of the cylinders. The showman hides a pound coin under one of the cylinders while the punter isn't looking. The punter places a pound coin bet by one of the cylinders. The showman lifts one of the other cylinders (that he knows isn't over the hidden coin) to reveal that the coin isn't under that cylinder, and offers the punter a chance to swap to the other remaining cylinder by doubling their bet. Once the punter has decided to double and move, or to stick with the original choice, both remaining cylinders are lifted. If the punter's coins are by the correct cylinder, he keeps the coins on the table, otherwise the showman takes them.

You decide to have a go. You place your bet against the right-hand cylinder. The showman lifts the centre cylinder to reveal nothing. You now have to choose whether to double your stake and swap to the left hand cylinder, or leave your bet where it is.

Question 1: On average, are you going to be better off moving your bet and doubling it, staying put, or does it not make any difference which you do?

As it happens, this time you stay put, and the showman lifts the cylinders to reveal that the hidden coin was indeed under the right-hand cylinder. You've won. The showman sets up the game again. This time, just after you've placed your bet on the middle cylinder, someone stumbles against the table, and the left-hand cylinder falls over, revealing —— nothing. The showman offers to continue the game anyway, as usual also offering you the choice of staying put, or moving (this time to the right-hand cylinder) and doubling your stake.

Question 2: Should you agree to continue with this game after this accident? (Yes, it was really a genuine accident. It wasn't some nefarious plan by the showman and an accomplice. "Bazaar of Honest Conmen", remember?)

Question 3: If you do continue with this game, are you more likely to be better off moving your bet and doubling it, staying put, or does it not make any difference which you do?


As always, answers in spoiler tags please. And an explanation of your answers would be appreciated. After all, you've a one in eighteen chance of getting them all right just by chance!