Problem 4. In a singing competition, a Rooster, a Crow, and a Cuckoo participated. Each member of the jury voted for one of the three performers. The Woodpecker calculated that there were 59 judges in total, and that the Rooster and the Crow received a total of 15 votes, the Crow and the Cuckoo received 18 votes, and the Cuckoo and the Rooster received 20 votes. The Woodpecker is a poor counter, but each of the four numbers he mentioned differs from the correct one by no more than 13. How many judges voted for the Crow? [6 points] (I.V. Raskina)
Answer: 13 judges.

Solution. The number of votes for the Rooster and the Raven cannot be more than $15+13=28$. Similarly, the number of votes for the Raven and the Cuckoo cannot exceed $18+13=31$, and the number of votes for the Cuckoo and the Rooster cannot exceed $20+13=33$. Adding these three quantities of votes, we get twice the total number of votes (each vote is counted in two of the three sums). Thus, the total number of jury members is no more than $(28+31+33) / 2=46$. On the other hand, from the first announcement of the Woodpecker, it is no less than $59-13=46$. Therefore, there are exactly 46 jury members, and all the inequalities are actually equalities.

Finally, the number of votes for the Raven can be found as the difference between the total number of jury members and the sum of those who voted for the Cuckoo and the Rooster: $46-33=13$ votes.