Say 16
Alice and Carla are playing a game often learned in elementary school known as Say 16 . The rules for the game are as follows:
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Each player takes turns saying between 1 and 3 consecutive numbers, with the first player starting with the number 1. For example, Player 1 could say the numbers 1 and 2, then Player 2 can say "3, 4, 5", then Player 1 can say "6" and so on.
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The goal of the game is to be the one to say "16".
Carla decides that she'll go first and that Alice will go second. Is there a way to tell which player is going to win before the game even starts?
Details and Assumptions :
- Assume that each player plays "perfectly", meaning that if there was an optimal way of playing, both players would be playing the best that the game allows them to play.
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The second player, Alice, has a winning strategy for this game.
No matter what Carla says, Alice should continue until the next multiple of 4 and then stop. For example, if Carla says "1, 2" Alice will say "3, 4." Then if Carla continues "5, 6, 7," Alice will just say "8." No matter what Carla then chooses (9, 9&10, or 9&10&11), Alice can always finish to the next multiple of 4, which is 12 in this case.
Finally, in the last turns, Carla will could say "13" or "13, 14" or "13, 14, 15," but no matter what she does, Alice will be able to say "14, 15, 16" or "15 16" or just "16" (respectively, given Carla's choice the previous turn) and so Alice will win.