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I will be using '3=' or 'X=Y=Z' to indicate a weighing like example 1 in the problem, '2>' or '(X & Y)>Z' to indicate a weighing like examples 2 and 3 in the problem, '1>' or 'X>(Y = Z)' to indicate a weighing like example 4 in the problem.
Group the nine coins into three groups of three. Weigh the coins in each group against each other. There are five possible scenarios:
Case 1: all three weighings are 3=
Case 2: one weighing is 3=, one weighing is 2>, and one weighing is 1>
Case 3: all three weighings are 2>
Case 4: two weighings are 2> and one weighing is 1>
Case 5: two weighings are 1> and one weighing is 2>
Case 1: all three weighings are 3=
The groups in some order are (20,20,20), (21,21,21), (22,22,22).
Label the coins in one group A's, in one group B's and the other group C's. Make a weighing with the three weights AAB/BBC/CCA.
Result Meaning
AAB<(BBC & CCA) The A group is 20g, the B group is 21g, the C group is 22g
AAB>(BBC = CCA) The A group is 22g, the B group is 21g, the C group is 20g
BBC<(AAB & CCA) The A group is 22g, the B group is 20g, the C group is 21g
BBC>(AAB = CCA) The A group is 20g, the B group is 22g, the C group is 21g
CCA<(AAB & BBC) The A group is 21g, the B group is 22g, the C group is 20g
CCA>(AAB = BBC) The A group is 21g, the B group is 20g, the C group is 22g
Total weighings: 4 (includes unneeded third 3= weighing)
Case 2: one weighing is 3=, one weighing is 2>, and one weighing is 1>
The 3= group is all the same weight, obviously. The two heavy coins in the 2> group plus the one heavy coin in the 1> group are all the same weight. The two light coins in the 1> group plus the one light coin in the 2> group are all the same weight. With this new grouping, the weighing described in Case 1 will finish sorting the coins.
Total weighings: 4
Case 3: all three weighings are 2>
The three light coins are the three 20g coins.
Take any three of the heavier coins and weigh them against each other.
If the result is 2> or 1> then the heavier coins in the weighing are 22g and lighter are 21g, repeat for the other three coins.
If the result is 3=, then the three coins are all 21g or 22g and the other three coins are the other of all 21g or 22g. Take one coin from one of the two groups and two coins from the other and weigh that set of three coins against each other. The lighter coins are from the group of 21g coins and the other group is the group of 22g coins.
Total weighings: 5
Case 4: two weighings are 2> and one weighing is 1>
One of the 2> groups is (22,22,21), the two light coins in the 1> group are two of the 20g coins.
Take the light coins from the two 2> weighings and weigh them against a known 20g. One of the coins will be the third 20g coin and the other will be a 21g coin. The two heavy coins from the group that the 21g coin was in weigh 22g. The remaining three unknown coins weigh 21g, 21g, and 22g. Use one weighing to determine which is which.
Total weighings: 5
Case 5: two weighings are 1> and one weighing is 2>
One of the 1> groups is (22,21,21), the light coin in the 1> group is 20g. Take one light coin from each of the two 1> weighings and weigh them against the known 20g coin. One coin will be the second 20g coin and the other light coin in the group it was from will be the third. The other coin will be 21g, the other light coin in the group the 21g coin was from will be the second 21g coin and the heavy coin in the group is a 22g coin. The remaining three unknown coins weigh 21g, 22g, 22g. Use one weighing to determine which is which.
Total weighings: 5
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