[pdurrant]

A variant on a classic problem.



I have twelve apparently identical coins. I keep them safe in a small bag, tucked away under my jacket. Well, in fact in two bags — so that even if one bag gets stolen, I won't have lost all my money (eggs and baskets, you know). The bags are not very pretty — just mass-produced things — but they are strong enough to hold the coins securely.

I suspect that one of my coins is a fake, and is either slightly heavier or slightly lighter than the others, and I need nine good coins to buy something I need to get today. Passing fake money is severely punished, so I can't just take a chance by picking nine of my twelve at random.

Unfortunately, the only scales available to me are owned by a miser who charges one coin per weighing — and I can't risk paying him with a fake coin either!

His scales are simple balance scales, two pans in which coins can be placed, and either the pans balance, or one side drops while the other rises.

Oh — and no-one's willing to lend me a known good coin or coins to weigh against my coins. A very unfriendly place, this.

Finally, I don't even have a pencil or paper. I'm going to have to work this out without writing anything down...

How can I end up with nine good coins from my twelve, and at the same time manage to pay the miser with good coins for however many times I need to use his scales? No matter what I do, I obviously can't afford to use his scales more than twice...





If you post a solution, don't forget spoiler tags. Thanks!