3. There is a pile of 100 matches. Petya and Vasya take turns, starting with Petya. Petya can take one, three, or four matches on his turn. Vasya can take one, two, or three matches on his turn. The player who cannot make a move loses. Which of the players, Petya or Vasya, can win regardless of the opponent's play?
Solution. Answer: only Vasya can win. Vasya, for example, can always make the remainder of the division by 3 of the number of matches left in the pile equal to 2 after his move. If there are exactly 2 matches left in the pile, then Petya takes one and loses on the next move. If there are 5 or more matches left, then Vasya has not made the last move yet (Petya will take no more than 4 matches on his next move).