6.3. At a round table sit 10 elves, each with a basket of nuts in front of them. Each was asked, "How many nuts do your two neighbors have together?" and, going around the circle, the answers received were 110, 120, 130, 140, 150, 160, 170, 180, 190, and 200. How many nuts does the elf who answered 160 have?
Answer: 55. Solution. We will call the elves 1-m, 2-m, etc., in the order of receiving answers. The odd-numbered ones sit every other. In their answers, the number of nuts of each even number is counted twice, so the sum $110+130+150+170+190=750$ is equal to twice the number of nuts of all even numbers. Therefore, the even-numbered ones have 375 nuts in total. The 8th and 10th together have 190 nuts, and the 2nd and 4th together have 130 nuts. Therefore, the 6th has 375-190-130=55 nuts.

Criterion. 1 point for the correct answer without justification. 3 points for the answer and a correct example for all.