Collection of counterfeit coin puzzles
We know that one of 30 coins, all of which look alike, is counterfeit and weighs a little less than a true coin. With the help of a balance scale (two pans; no weights), can the counterfeit coin always be found in three weighings?
The counterfeit coin cannot be found among 30 coins in three weighings. For in the first weighing either at least 10 coins are on each pan or at least 10 coins are left off the weighing, but either way there is a set of 10 coins that could contain the counterfeit. In the second weighing, either at least 4 coins are on each pan or at least 4 coins are left off. The counterfeit could be in this set of 4. In the third weighing, either at least 2 coins are on each pan or 2 coins are left off. The counterfeit could be in this set of 2, so that after three weighings, we haven't succeeded. The counterfeit can be found in the case of 27 (or fewer) coins. To see how, first divide the coins into three groups of nine coins. In the fust weighing, compare two of the groups. If it balances, the counterfeit coin is in the third group, otherwise it is in the lighter group. Divide the group with the counterfeit coin into three groups of three coins and proceed with the second weighing in the same way as the fust. In this way you will find the group which contains the counterfeit coin. Now you can take two of the three coins and compare them. If they balance the counterfeit is the third coin, otherwise it is the lighter coin. Generalizing, given \(n\) weighings we can find the counterfeit amongst \(3^n\) coins.
There are 101 coins, out of which, 1 is fake. The fake coin is identical to a genuine coin, except that it differs in weight. Using a balance scale only twice, how can you determine whether the fake coin is heavier or lighter than a genuine coin? source/solution
There are 101 coins, 51 genuine, 50 fake. Every genuine coin has the same weight. For fake coins, there are two possibilities:
- Each fake coin is 1 unit lighter than each genuine coin.
- Each fake coin is 1 unit heavier than each genuine coin.
You are given 1 of the 101 coins, and must determine whether it is genuine. You are given a balance scale that tells you the difference in weight between its left and right pans. You can weigh any of the 101 coins, but you can only use the balance scale once.
A bunch more problems like this are available from JRMF.