8.3 There are twenty-one visually identical coins, one of which weighs 9.9 g, two others weigh 9.8 g each, and all the rest weigh 10 g. Can at least one 10-gram coin be identified in two weighings using a balance scale without weights?
Answer: Yes.

Solution: Let's denote the coins by numbers from 1 to 20. Weigh coins 1 and 2 against coins 3 and 4. Among these four coins, there is at least one genuine coin.

In the case of balance, each pan either has two genuine coins or one genuine and one 9.8 g coin. These cases can be distinguished by the second weighing: weigh coins 1 and 2.

If one pan outweighs the other in the first weighing, then the heavier pan contains a genuine coin. Weigh the two coins from the heavier pan against each other. If they balance, both are genuine; if they do not balance, choose the heavier coin.

Criteria: Incorrect algorithm or omission of more than one case - 0 points.

Omission of one case - no more than 3 points.

Algorithm of weighing only, without justification - no less than 5 points.