72*. Two players play the following game. There is a board on which the number 1000 is written, and a pile of 1000 matches. On each turn, each player (they take turns) can either take from the pile,
or add no more than five matches to it (initially, neither player has any matches), and then the number of matches in the pile after this move is written on the board. The player who, after their move, writes a number that has already appeared on the board loses. Who wins with correct play: the first player or their opponent?
89.72. Answer. The second player wins. He mentally divides all numbers from 0 to 999 into 125 sequences of eight consecutive numbers, and then acts as follows: after the first player writes the first (not necessarily the largest) number in any of these sequences on the board, the second player makes a countermove by writing the smallest possible number of the same sequence on the board (taking three, four, or five matches). The remaining six matches the second player mentally divides into three pairs (in order), and for each move of the first player, which consists of writing one of these numbers, the second player responds by writing the other number of the same pair; it is easy to verify that the matches taken by the second player on the first move within this sequence are always sufficient for this (it is assumed that these matches the second player sets aside in a separate pile and uses them only for the game within this sequence). Thus, the second player always has a response and, consequently, he wins.