Answer to the Coin-Weighing Problem

1
0
0
+
2
0
+
-
3
0
+
0
4
0
+
+
5
+
-
-
6
+
-
0
7
+
-
+
8
+
0
-
9
+
0
0
10
+
0
+
11
+
+
-
12
+
+
0
13
+
+
+
To the left is a box with each of the numbers one to thirteen (one row for each coin) expressed in the zero-centered base three. The columns are the nine's, three's, and one's places. Each column determines how to position the coins for one of the weighings on the scale-zero for coins not being weighed, and + and - indicating on which side of the scale to put a coin. The fourteenth coin is used as necessary to equalize the number of coins on each side of the scale.
The trick here is to use one rule for odd-numbered coins and a different rule for even-numbered coins. For odd-numbered coins let + indicate the right-hand side of the scale and - the left-hand side. Reverse this rule for even-numbered coins. For the first weighing, then, coins 5, 7, 9, 11 and 13 would go on the right side of the scale, and coins 6, 8, 10, 12 and 14 (so that there are 5 coins on each side) would go on the left. For the second weighing, coins 3, 6, 11, 13 and 14 would go on the right, and coins 2, 4, 5, 7, and 12 would go on the left.
To convert the results of the three weighings into an answer to the puzzle, one takes the first weighing to have a value of +9 (if the right side is heavier) or -9 (if the left side is heavier). The second weighing would have a value +3 or -3, and the third weighting a value of +1 or -1. Add the results. If zero, no coins are counterfeit. For non-zero results, the absolute value of the answer indicates which coin is off. A negative result indicates a light odd-numbered coin or a heavy even-numbered coin. A positive result indicates the opposite.