Question 134
Arnaldo and Bernaldo decide to play a game that has a limited number of moves. Arnaldo writes the number 1 on the board in his first move. Then, Bernaldo writes 2 or 4 on the board. After that, Arnaldo writes 3 or 9 on the board. The two keep playing alternately, following the rule that in move $n$ the player writes $n$ or $n^{2}$ on the board. Arnaldo wins the game if, after the last move, the sum of the numbers on the board is divisible by 3. If the sum is not divisible by 3, then Bernaldo wins.
a) Suppose the game ends at move number 15. Show that Bernaldo can guarantee victory.
b) Suppose the game ends at move number 7. In this case, which of the two players can always guarantee victory regardless of how his opponent plays? How should he play to win?
a) Suppose the game ends at move number 15. Show that Bernaldo can guarantee victory.
b) Suppose the game ends at move number 7. In this case, which of the two players can always guarantee victory regardless of how his opponent plays? How should he play to win?