Theory
A balance puzzle or weighing puzzle is a logic puzzle about balancing items—often coins—to determine which holds a different value, by using balance scales a limited number of times. These differ from puzzles that assign weights to items, in that only the relative mass of these items is relevant.
Problems
1. There are 3 coins. All of them look the same. One coin is false. Its weight is less
than the weight of a true coin. All true coins weigh the same number of grams. You are
allowed to use a set of balance scales. Is it possible to isolate the counterfeit coin with
only one weighing?
2. Now you have 9 coins. All of them look the same. One of them is false. Its weight
is less than the weight of a true coin. All true coins weigh the same number of grams.
You can use a set of balance scales. Is it possible to isolate the counterfeit coin with
only two weighings?
3. 5 peaches are lying on the table. We want to know their total weight. But
unfortunately, it is possible to put exactly two peaches on the weighing scale simultaneously.
(a) How can we measure the total weight of these peaches? What is the least number
of weighings we need to perform? (b) What is the least number of weighings we need
to determine the total weight of 13 peaches?
4. Let's go over the coin problems with larger values. Assume now you have n coins. Again
all the coins look the same and there is only one false coin, but now it is heavier than
the true coin. The true coins weigh the same amount of grams. You are allowed to
perform measurements with the help of a set of balance scales.
(a) Can you isolate the counterfeit in 3 weighings if n = 27?
(b) Isolate the counterfeit in 4 weighings for n = 81.
(c) What is the smallest number of weighings you need to perform to determine the false coin for n = 82 and n = 80?
5. Generalize the result of Problems 1, 2, and 4.
6. Now you are given 6 precious stones. It is known that there are 2 counterfeit stones
of equal mass among them, and each of the other 4 real stones weighs more than each of
the counterfeits. Once again we have a set of balance scales, but we cannot make only
3 weighings to isolate both counterfeit stones. How can we do it?
7. One hundred and one coins were produced. Something went wrong after minting
the first 51 coins and the last 50 coins are defective. The computer shows that the defective
coin should weigh less than the true by 1 gram. We know that all the true coins weigh
the same integer number of grams, and so do the false coins. Suppose you have picked a
random coin out of these 101 coins. You have a set of balance scales which can show
you the difference (in grams) between the weights of the objects on the pans. You need to
perform one weighing to understand whether it is real or fake.