11.4. On each of 2013 cards, a number is written, and all these 2013 numbers are distinct. The cards are face down. In one move, it is allowed to point to ten cards, and in response, one of the numbers written on them will be reported (it is unknown which one). For what largest $t$ can it be guaranteed to find $t$ cards, for which it is known what number is written on each of them?

(I. Bogdanov)
Answer. $t=1986=2013-27$.

Solution. 1. First, we will show that it is impossible to guess 1987 cards. Number the cards $A_{1}, \ldots, A_{2013}$; we will demonstrate how to arrange the answers so that none of the numbers on the cards $A_{1}, \ldots, A_{27}$ can be determined.

For each $i=1, \ldots, 9$, combine the cards $A_{3 i-2}, A_{3 i-1}, A_{3 i}$ into a triplet $T_{i}$. If among the specified 10 cards there is a card $A_{n}$ with $n>27$, we will answer the number $n$. If all 10 cards lie within the triplets, then in some triplet $T_{i}$ there are at least two cards; in this case, we will answer the number on the edge between these cards as shown in Fig. 8. These answers are satisfied by two scenarios: the numbers on the cards with numbers greater than 27 are their own numbers, and either on each card in the triplets, the number on the edge coming out of it counterclockwise is written, or on each such card, the number on the edge clockwise is written. Therefore, none of the numbers on the cards in the triplets can be determined.

![](https://cdn.mathpix.com/cropped/2024_05_06_6370f76bc3967a655918g-3.jpg?height=307&width=323&top_left_y=895&top_left_x=968)