On the table are $300$ coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take $1, 2, 3$, or $4$ coins from the table, Vasya, $1$ or $2$ coins, and Tolya, too, $1$ or $2$ coins. Can Vasya and Tolya agree so that, as if Petya were playing, one of them two will take the last coin off the table?
Problem
Source: Caucasus 2015 8.5
Tags: combinatorics, game strategy, game
natmath
26.04.2019 16:31
On the table are $300$ coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take $1, 2, 3$, or $4$ coins from the table, Vasya, $1$ or $2$ coins, and Tolya can also take $1$ or $2$ coins. Can Vasya and Tolya agree so that, as if Petya were playing one of Vasya or Tolya will take the last coin off the table? I think that means that can Vasya and Tolya team up and take the last coin off the table to win
celilcelil
06.03.2021 21:23
Yes, I also think that. Vasya and Tolya can't do it. First Petya take 1 coin. Now there are 299 coins which is 5 (mod 6). If Vasya and Tolya take togather 2 coins, then Petya takes 4 coins, if they take 3 coins, then Petya takes 3 coins, if they take 4 coins, then Petya takes 2 coins. In this strategy there are always a number which is 5 (mod 6) before Vasya plays. Then after 148 move there will be 5 coins and it will be Vasya's move. If they take 2 coins togather, Petya will take 3 coins,if they take 3 coins, Petya will take 2 coins, if they take 4 coins, Petya will take 1 coins and Vasya and Tolya can't win.