10.8. A numismatist has 100 coins that look identical. He knows that among them, 30 are genuine and 70 are counterfeit. Moreover, he knows that the weights of all genuine coins are the same, while the weights of all counterfeit coins are different, and any counterfeit coin is heavier than a genuine one; however, the exact weights of the coins are unknown. There are balance scales without weights, on which it is possible to compare the weights of two groups consisting of the same number of coins in one weighing. What is the minimum number of weighings on these scales that the numismatist can guarantee to find at least one genuine coin?

(C. Berlov, I. Bogdanov)
Answer: 70.

Solution: 1. We will show that the numismatist can find the genuine coin in 70 weighings. Stack all 100 coins in a pile. With each weighing, he will select two coins from the pile and compare them. If their weights are equal, then both coins are genuine, and the required coin is found. If not, then the heavier coin is counterfeit, and it can be discarded from the pile.

After 70 such weighings, if equality is never achieved, then 30 coins will remain in the pile, and all genuine coins will remain in the pile. Thus, in this case, the numismatist will even find all 30 genuine coins.