1. [4] The magician lays out a deck of 52 cards in a row and announces that 51 of them will be thrown off the table, leaving only the three of clubs. On each step, the spectator says which card to discard by its position from the edge, and the magician chooses whether to count from the left or the right edge and discards the corresponding card. For which initial positions of the three of clubs can the trick be guaranteed to succeed?

(Alexey Voronaev)
Answer: at the extreme positions. Solution. The three of clubs will have to be discarded only if it ends up in the center of the row at some point; otherwise, another card can be discarded. Since the row always contains more than one card, an end card can be kept until the end.

Let the three of clubs $T$ initially not be at the end. We will present two strategies for the audience.

Strategy 1. The audience names any number except the extreme numbers, preventing the removal of the end cards. When only three cards remain, the three of clubs (if it is still on the table) will be in the center. The audience will name 2.

Strategy 2. Suppose the audience always guesses the position number of $T$. The magician will have to discard another card (he has no choice), reducing by one the greater of the distances from the three of clubs to the edge. Thus, eventually the distances to the edges will coincide, and the three of clubs will have to be discarded.