Race to 32.768!

Logic Level 3

Alice and Bob play a game with the following rules:

  • There are 9 cards on a table with the numbers: 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 , 512 2,4,8,16,32,64,128,256,512
  • In each turn a player pick exactly one card.
  • Whoever gathers 3 cards with a product of 32768 32768 wins.
  • Both players play optimally.
  • Alice starts first.

Question: Which player has a winning strategy?

Neither Alice Both Bob

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1 solution

Steven Yuan
Jun 7, 2017

This is basically a spin on the classic "sum to 15" problem, except it's with exponents and products.

Notice that 32768 = 2 15 32768 = 2^{15} and that the cards have numbers of the form 2 k , 2^k, where 1 k 9. 1 \leq k \leq 9. When we multiply the values in any two of the cards, we are adding their exponents together.

Thus, Alice and Bob are trying to pick up cards such that the sum of their exponents equals 15. This can be shown to correspond to a game of Tic-Tac-Toe, with a grid of numbers from 1 to 9 such that the sum of each row, column, and long diagonal is equal to 15. Since Tic-Tac-Toe, when played optimally, is a draw, we conclude that neither Alice nor Bob has a winning strategy.

Great solution!

maximos stratis - 4 years ago

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