57 (980). Two people play such a game: the first one calls out any integer from 1 to 10 inclusive, the second one mentally adds another number from the first ten to this number and calls out the sum. To this sum, the first one adds again some number from the first ten and calls out the sum, and so on. The one who calls out the number 100 wins. How should one play to win? Who wins with the correct play: the one who starts, or the partner?
Solution. To win (by naming the number 100), one needs to force the opponent to name a natural number from the interval $[90 ; 99]$. For this, one needs to name the number 89 oneself. To be able to name the number 89 oneself, one needs to force the opponent to name some natural number from the interval $[79 ; 88]$, for which one needs to name the number 78 oneself. Reasoning similarly, we find that with correct play, the one who starts can always win if they name the following numbers: 1, 12, $23, 34, 45, 56, 67, 78, 89$. If the one who starts names any other number, the partner wins by naming a number from the specified sequence. (This number, according to the rules of the game, is obtained by adding some number from the first ten to the number named by the one who starts.)