You have four coins to sort with a standard balance scale. Their weights are 20g, 21g, 22g and 23g. Prove that there is no strategy which can guarantee sorting the coins with only three weighings.
Call the four coins A, B, C, D.
Weigh A+B vs C+D; A+C vs B+D; A+D vs B+C; on two cases there will be no balance (22+23 vs 20+21, 21+23 vs 20+22) and in the last one there will be balance (20+23 vs 21+22).
The coin on the heavier side on both unbalanced weighings is the 23g one.
The coin on the lighter side on both unbalanced weighings is the 20g one.
Now, with this method... which is the 21g one and which the 22g one?
Of course, this isn't a proof; just a method that didn't work...