__(Easy) Four weighing solution__:
Weigh 8 vs 8. If unequal keep the heavier side, if equal keep either side.
Split that side into halves and weigh 4 vs 4. Again, if unequal keep the heavier side, if equal keep either side.
Repeat for 2 vs 2 and 1 vs 1. The single coin picked at the end will be from the heavy set.
__(Hard) Three weighing solution__:
Weigh 7 vs 7 with 2 coins left off.
Case 1: 7 vs 7 is unequal.
- The heavy side has at least 4 heavier coins, Weigh 3 vs 3 from that side. If unequal the heavier side has at least two 31 g coins, take that side. If equal both sides have at least two heavy coins, take either side.
- Make the last weighing 1 vs 1 from the group of three chosen. Either the weighing is unequal which gives a heavy and light coin or the weighing is equal which means both are heavy.
Case 2: 7 vs 7 is equal.
- Both sides contain 3 light and 4 heavy or both contain 4 light and 3 heavy, the two coins left off must be of equal weight. Choose one side and weigh 3 vs 3 with the last coin left off.
Subcase 2-1: 3 vs 3 is unequal.
- The heavy side must have at least two heavy coins. Make the third weighing 1 vs 1 similar to Case 1.
Subcase 2-2: 3 vs 3 is equal.
- Both sides contain 1 light and 2 heavy or both contain 2 light and 1 heavy. Furthermore the coin left off the second weighing must weigh the same as the two coins left off in the first weighing. Make a set of three from these left off coins and compare that set to one of the sets of three from the second weighing. That weighing will determine what weight the three previously left off coins weigh. |