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You are given nine coins, one of which is fake and is either lighter or heavier than the others. You have a balance scale that you can use only twice. How can you identify the fake coin and determine whether it is lighter or heavier than the others using only two weighings?
1. Divide the nine coins into three groups of three coins each, and place them on the balance scale as shown below:
A B C
———- ———- ———
1 2 3 4 5 6 7 8 9
2. Weigh the first two groups (A and B) against each other. There are three possible outcomes:
– If A is lighter than B, then the fake coin is in group A, and it is either a lighter coin or the only normal weight coin in that group.
– If A is heavier than B, then the fake coin is in group A, and it is either a heavier coin or the only normal weight coin in that group.
– If A and B balance, then the fake coin is in group C.
3. Now that you know which group the fake coin is in, divide that group into three individual coins, and place one on each side of the balance scale, with the third coin aside. For example, if the fake coin is in group A, place coins 1 and 2 on one side of the balance scale, and coin 3 aside.
4. There are again three possible outcomes:
– If the two coins on the balance scale balance, then the fake coin is the third coin aside, and you know that it is a lighter or heavier coin.
– If the two coins on the balance scale do not balance, and the side with two coins is heavier, then one of those two coins is the fake coin, and it is heavier than the normal weight coins. The third coin aside is a normal weight coin.
– If the two coins on the balance scale do not balance, and the side with two coins is lighter, then one of those two coins is the fake coin, and it is lighter than the normal weight coins. The third coin aside is a normal weight coin.
By following these steps, you can identify the fake coin and determine whether it is lighter or heavier than the others, using only two weighings.