David G. Cantor and W. H. Mills,
"Determining a Set from Certain Combinatorial Properties,"
Canad. J. Math., 18, 1965, pp. 42-48.
Weigh three coins, say the first, second and fourth. Then weigh the
first and third. Then weigh the second and third. The sum of the results
of all three weighings is twice the weight of first, second and third
coins together, but only once the weight of the fourth coin. So from the
addition one knows whether the fourth coin is good or bad (because the
weights of good and bad coins are known and all coins have one or the
other of these weights). Then one knows (first weighing, subtract off
the fourth coin's weight) the weight of coins one and two together. By
adding the last two weighings, subtracting this coin one and two total,
one obtains twice, and then exactly, the weight of the third coin. From
that value and each of the latter two weighings by subtraction one
learns the weights of coins two and three.