There are 14 coins, numerated from 1 to 14, and presented as an evidence at the court. The first 7 are fake coins, and others are genuine coins. Fake coins have all the same weight and they are lighter than the genuine coins. How can an expert prove that the first 7 coins are fake, and that other coins are genuine, using balance, in at most 3 weighings? ( We don't know whether the genuine coins have the same weight, but if someone can solve problem in case when they have the same weight, I would appreciate it.)
Solution where all fake coins weigh the same and all regular coins weigh the same (and all fake are lighter). Label the fakes ABCDEFG, regulars abcdefg.
Test A vs. a. See that A < a, and therefore A is fake and a is real.
Test aBC vs. Abc. See that aBC < Abc. Since A > a, we have 1 unit of difference going to the left, so it must take at least two units of difference the other way to make the scale tilt to the right. Thus, it must be the case that both BC are fake and both bc are real.
Test abcDEFG vs. ABCdefg. See that abcDEFG < ABCdefg. As above, this proves that ABCDEFG are all fake and abcdefg are all real.
If the real coins could vary in weight, then you could never conclude anything by weighing coins and seeing they are different, since that could mean they are all real.