I was intending to give the most-memorable solution I know to the 12 coins problem. It was published in a poem in "Eureka", the Cambridge University undergraduate mathematical journal, many years ago. I the poem, the problem of finding the fake coin was posed to Felix by his mother. He labels the coins with the letters of F. AM NOT LICKED and then performs the weighings indicated by
MA DO LIKE ME TO FIND FAKE COIN.
It's easy to check that the 25 different cases (whether there is a fake coin, and if so which it is, and whether it's heavy or light) all give different answers. This solution is due to C.A.B.Smith, writing under a pseudonym I've forgotten.
In the general case n weighings suffice to locate a fake coin out of (3^n - 3)/2, and it's impossible to do better even though the "information count" would allow (3^n - 1)/2. Just the other day I read a very neat solution in a popular book by Dan Pedoe, which is in my office so I could retail it tomorrow if anyone would like to see it. [It's distinctly more elegant than anyone could reasonably expect to find in a week.]