10.5. The tsar has eight sons, and they are all fools. Every night the tsar sends three of them to guard the golden apples from the firebird. The tsareviches cannot catch the firebird, blame each other for it, and therefore no two of them agree to go on guard duty a second time. For how many nights can this continue at most?
Answer: 8 nights.

Solution. Evaluation. Consider any of the sons. In each guard, he is with two brothers, so after his three appearances, there will be one brother with whom he has not been on guard and will not be able to go, as there will be no third for them. This situation is "symmetric," meaning if son A has not been on guard with son B after his three appearances, then son B has also not been on guard with son A after his three appearances. Therefore, there are at least four pairs of sons who have not and will not be in the same guard. The total number of possible pairs is $\frac{8 \cdot 7}{2}=28$, so there can be no more than $28-4=24$ pairs in the guards. Since three pairs of sons are involved each night, the number of nights can be no more than $24: 3=8$.

Example. Let the sons be denoted as: 1, 2, 3, 4, A, B, C, D. Then eight triplets satisfying the problem's condition are: A12, B23, C34, D41, AB4, BC1, CD2, DA3.

An example can also be provided in the form of a diagram (see Fig. 10.5). Eight sons are represented by points, and eight triplets satisfying the problem's condition are represented by lines.

Grading criteria.

“+” A complete and justified solution is provided.
“士” A generally correct reasoning is provided, containing minor gaps or inaccuracies, and the correct answer is obtained.

“干” The correct answer and evaluation are provided, but the example is missing or incorrect.
“Ғ” The correct answer and example are provided, but the evaluation is missing or incorrect.

“-” Only the answer is provided.
“-” The problem is not solved or is solved incorrectly.