Coin Parity

Konstantin Knop wrote to me about a weighing puzzle. Apparently, the balance scale keeps inserting itself into our conversation. Puzzle. There are several coins, each weighing either 10 grams or 11 grams. Determine the parity of the number of coins of each weight using a balance scale. Given the total number of coins, it is enough to find one of the parities. The other parity follows. The first interesting case has four coins. How can we determine the parity of the number of 10-gram coins in two weighings? The solution is on the surface: compare the coins in pairs. A balanced pair contributes either zero or two 10-gram coins, so it does not change the parity. An unbalanced pair contributes exactly one 10-gram coin, so it changes the parity. Thus, the parity is the parity of the number of unbalanced weighings. Now try eight coins. The same pairwise method takes four weighings. But three weighings are enough. Puzzle. There are 8 coins, each weighing either 10 grams or 11 grams.…