2. There are 25 coins, 12 of which are counterfeit and differ in weight by exactly 1 g from the genuine ones. All coins weigh an integer number of grams. Some may be lighter than the genuine ones, while others may be heavier. There are balance scales without weights, with a needle that shows the difference in weight. What is the minimum number of weighings needed to determine whether a given coin is counterfeit or not?
Solution. We will prove that one weighing is sufficient. Set aside the coin under investigation, and place the rest on the scales, 12 coins on each side. If the scales show a difference in an even number of grams, then the coin is genuine; if in an odd number, then it is counterfeit. Indeed, if an odd number of counterfeit coins lie on the pan, then its mass differs from the mass of 12 genuine coins by an odd number of grams (the sum of an odd number of odd numbers), otherwise - by an even number. Depending on the coin under investigation, there are 11 or 12 counterfeit coins on the pans, which can be divided into two groups with different or the same parity of the number of coins, respectively. Answer: one weighing.