There are 6 coins weighing 1, 2, 3, 4, 5 and 6 grams that look the same, except for their labels. The labels are supposed to display the weights of the coins.
How can you determine whether all the labels are correct,
using the balance scale only twice?
There are only two sums of the form A+B+C=D+E in the set {1,2,3,4,5,6}. Those are 1+2+6=4+5 and 2+3+5=4+6.
Make the two weighings reflecting those sums: coins labeled {1,2,6} vs {4,5} and coins labeled {2,3,5} vs {4,6}. Both weighings will balance only when all the labels are correct.
Solution
