부산대학교 도서관

In a collection of coins there may be some that are different. We use a two-pan balance to study when they can be identified. We first establish some terminology. We do not use the adjectives real, fake, genuine or counterfeit as they all have irrelevant connotations. Instead, we have normal coins all of which have the same weight. There may be abnormal coins each heavier than a normal coin. There may also be abnormal coins each lighter than a normal coin. Any difference in weight is insignificant compared to the weight of a normal coin. There is no need to say that the coins have identical appearance. If each has a unique appearance instead, the problem and the solution will not be affected. We now establish some notation. We use capital letters to label the coins. We use a two-pan balance which only indicates which pan, if either, contains coins with a higher total weight. We always put the same number of coins in each pan. Coins in the same pan are enclosed in brackets. The brackets are omitted if the number of coins in the pan is one. When (A,B) is compared with (C,D), (A,B)=(C,D) means that we have equilibrium; (A,B)>(C,D) means that the total weight of A and B is higher than the total weight of C and D; (A,B)<(C,D) means just the opposite. Problem 1. All coins are normal except for possibly two, a heavy coin and a light coin whose total weight is equal to the total weight of two normal coins. Using at most four weighings, determine whether two such abnormal coins exist, and to identify them if they do. The total number of coins is (a) 5; (b) 7; (c) 9. Problem 2. The total number of coins is at least 4. All are normal except for a heavy coin and a light coin. Using at most four weighings, determine whether the total weight of the two abnormal coins is (1) greater than, (2) equal to, (3) less than the total weight of two normal coins. [ABSTRACT FROM AUTHOR]