# 1.1. Condition:

Twelve figures are made of matches - 3 triangles, 4 squares, and 5 pentagons. The figures have no common sides. Petya and Vasya take turns removing one match at a time. Vasya wants to leave as few untouched figures as possible, while Petya wants to leave as many untouched figures as possible. How many figures will remain after 10 moves? Each of the boys makes 5 moves, with Petya starting first.
Answer: 6

Solution. Vasya will be able to take matches from a maximum of 5 different figures. This means that if Petya takes matches only from those figures that have already lost at least one match, then $12-5=7$ figures will remain untouched. Petya can fulfill the condition, as otherwise, Vasya would have taken a match from a partially dismantled figure. However, at the beginning, Petya will have to take a match from an untouched figure.

Note that each figure has at least three matches, so Petya will always be able to make his move, and since there are 12 figures, Vasya will also be able to make his moves. Thus, the boys will be able to adhere to their strategies.