# Six Misnamed Coins, Two Weighings

One of the problems at the last round of LIV Moscow Mathematical Olympiad

There are 6 coins weighing 1, 2, 3, 4, 5 and 6 grams that look the same, except for their labels. The labels are supposed to display the weights of the coins. How can one determine whether all the labels are correct, using the balance scale only twice?

The problem is due to Sergey Tokarev. Tanya Khovanova and Joel Lewis extended the problem to any number of coins.

Solution

### References

1. Kvant, 1991, issue 9, pp. 70-71 (in Russian) There are 6 coins weighing 1, 2, 3, 4, 5 and 6 grams that look the same, except for their labels. The labels are supposed to display the weights of the coins. How can one determine whether all the labels are correct, using the balance scale only twice?

### Solution

First weigh the coins labeled 1, 2, 3 against the one labeled 6. In the absence of balance the problem is solved in 1 weighing. The only time when the weights may be equal is when 6 is labeled correctly. But not only that. If the first weighing shows a balance, the coins labeled 1, 2, 3 may only be misnamed among themselves, and so are the coins labeled 4, 5.

So assume that and move to a second weighing. Weigh coins 1, 6 against coins 3, 5. If the 4 labels are correct then all labels are correct and the pair {3, 5} outweighs the pair {1, 6}. This is the only case where that may happen. Indeed, mislabeling in the groups {1, 2, 3} and {4, 5} may only result in the weight of the pair {3, 5} to go down and the weight of the pair {1, 6} to go up.

Therefore, if on the second weighing the pair {3, 5} outweighs the pair {1, 6}, all coins are labeled correctly. Otherwise, some are not. ### Weighing Coins, Balls, What Not ... 