Posted June 4, 2017 By Presh Talwalkar. Read about me , or email me .

On a table are nine face-up cards. Each card displays a different number from the list: 2, 4, 8, 16, 32, 64, 128, 256, 512.

Alice and Bob alternately take turns. Each picks a single card on a turn, and Alice goes first. Once a card is chosen it cannot be selected again. The game ends when someone wins or all the cards have been selected.

The winner is the first person to collect a set of 3 cards with a product of 32,768. If neither player does this, the game ends in a draw.

If both play optimally, does either player have a winning strategy? Or does the game always end in a draw? How should you play this game?

Can You Solve The Race To 32,768 Puzzle?

Or keep reading.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Answer To The Race To 32,768

The game sounds complicated, but it is simple to understand once you know the trick!

The game involves the powers of 2 from 1 to 9. The goal is to collect a set of 3 numbers whose product is 32,768 = 215.

We can transform the game from multiplication into addition by taking the base 2 logarithm of each number. In that game, Alice and Bob pick numbers between 1 and 9. Instead of having 3 numbers multiply to 215, the goal is to collect 3 numbers that add up to 15.

We can visualize this game using a magic square, in which every row, column, and the two diagonals sum to 15:

4 9 2 3 5 7 8 1 6

This is a 3×3 magic square, and it is a unique arrangement (there are 8 patterns, but each is a rotation or reflection of the same pattern, which I proved here.)

Imagine Alice plays X on a square when picking a number and Bob plays O. Having 3 numbers that sum to 15 only happens when a player has 3 in a row. In other words, this game is equivalent to tic-tac-toe!

The multiplicative 3×3 square is found by writing 2x in each entry:

16 512 4 8 32 128 256 2 64

As everyone knows, the game of tic-tac-toe ends in a draw with optimal play. (See the best way to play tic-tac-toe).

Thus, if Alice and Bob play optimally, this multiplication game also ends in a draw.

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Source

I read about the 3×3 addition game in Peter Winkler’s book Mathematical Puzzles: A Connoisseur’s Collection.