There are many solutions to the puzzle. However, each solution will share four central elements. First, the coins are placed into three groups, each containing four coins. Second, the first weigh is one set of four coins against another set of four coins. Third, the information from the previous weighs is retained. And fourth, coins are eliminated by analyzing coins in groups of three.
As a specific example, consider that the coins are grouped in three sets and are labeled as ABCD, 1234, and abcd.
Let’s consider the case where the coin is one of abcd.
On the first weigh, compare ABCD against 1234. If they are equal then you know that the coin you seek is in abcd.
On the second weigh, compare ab against c1, setting d aside. You know that 1 is not the coin you seek based on the results of the first weigh (you could have chosen any coin from ABCD or 1234here). If they are equal, then you know the coin you are looking for is d. If they are not equal, then the coin is one of abc. Note the direction that the scale is tipping in and move to the third weigh.
On the third weigh, compare a versus b, by placing b on the other side of the scale and setting caside. If they are equal, the coin you seek is c. If they are not equal, but the scale tips in the same direction as the previous weigh, then the coin you seek is a. If they are not equal, but the scale tips in the other direction, then the coin you seek is b.
(Note: The case where the first weigh is equal can be solved without creating a group of three in the second weigh. I present the solution in this manner to illustrate similarities between this easier case and the more difficult case that follows. )
Now consider the case where the coin is one of ABCD or 1234.
On the first weigh, you compare ABCD against 1234. If they are not equal then you know that the coin you seek is in ABCD or in 1234. Note the direction that the scale is tipping in and move to the second weigh.
On the second weigh, compare AB12 against C3ab, by moving 12 and C to the other side, setting D and 4 aside, and adding ab to ensure you weigh the same number of coins on each side. (You know that the coin is not abcd from the first weigh). If equal, then the coin you seek is either D or 4. This can be solved on the third weigh by weighing D against a. If they are equal then the coin you seek is 4, otherwise it is D. If the second weigh of AB12 against C3ab was not equal, you need to determine if the scale is tipping in the other direction from the first weigh. If it is tipping in the same direction as the first weigh, then the coin you seek is in AB3. If it is tipping in the other direction than the first weigh, then the coin you seek is in C12. Note the direction that the scale is tipping in and perform the third weigh.
On the third weigh, if the coin is in AB3, compare A against B, moving B to the other side and setting 3 aside. If equal, then the coin you seek is 3. If it is not equal and tips in the same direction as the second weigh, then the coin you seek is A. If it is not equal and tips in the other direction as the second weigh, then the coin you seek is B.
On the third weigh, if the coin is in C12, compare 2 against 1, moving 2 to the other side and setting C aside. If equal, then the coin you seek is C. If it is not equal and tips in the same direction as the second weigh, then the coin you seek is 1. If it is not equal and tips in the other direction as the second weigh, then the coin you seek is 2.
Thus, you are able to find the coin with the different weight in three weighs.
