I saw the following coin weighing problem on Tanya Khovanova's blog.
Puzzle. At a trial, 14 coins were presented as material evidence. The expert tested the coins and discovered that seven of them were fake, the rest were real, and he knew exactly which coins were fake and which were real. The court only knows that counterfeit coins weigh the same, real coins weigh the same, and fake ones are lighter than real ones. The expert wants to use not more than three weighings on a balance scales without weights to prove to the court that all the counterfeit coins he found are really fake, and the rest are real. Could he do it?
In the solution below, a "known" coin is one whose identity is known to the court and an "unknown" one is one whose identity is not known to the court (but is still known to the expert).
On the first weighing, place a single unknown real coin on one scale and a single unknown counterfeit coin on the other scale. This determines one real and one counterfeit coin.
In the second weighing, place the known real coin and two unknown counterfeit coins on one scale and the known counterfeit coin and two unknown real coins on the other scale. The weighing will show that the scale containing the real coin is lighter and hence contains more counterfeit coins than the other scale. The only way this is possible is for the the two unknown coins on the lighter scale to be counterfeit and the two unknown coins on the heavier scale to be real. Thus we have 3 known real coins and 3 known counterfeit coins.
In the third and final weighing, place 3 known real coins and 4 unknown counterfeit coins on one scale and 3 known counterfeit and 4 unknown real coins on the other scale. The situation is analogous to the second weighing and allows us to deduce the identity of the remaining 8 unknown coins.
The procedure used in the second and third weighing generalizes to allow the expert to use N weighings to prove the identity of 2N - 1 real and 2^N-1 counterfeit coins.