12 coins, one fake of unknown weight
Asked at Jane Street, SIG, Optiver
You have 12 coins, identical in appearance. Exactly one is counterfeit and has a different weight than the rest, but you do not know whether it is heavier or lighter. You have a two-pan balance.
In just three weighings, identify the fake coin and determine whether it is heavy or light.
Show a hint
There are 24 possible answers (12 coins × heavy/light). Three weighings give outcomes. That's barely enough, so no weighing can be "wasted." Never leave the pans unbalanced when you could learn more.
Your answer
This one is open-ended. Work it through, then check your reasoning against the full solution.