Sum to Fifteen

Source(s):

• I found this puzzle in Mathematical Puzzles: Connoisseur’s Collection, Peter Winkler, which attributes it to

• Vol. II of Winning Ways for Your Mathematical Plays by Elwyn Berlekamp, John Conway and Richard Guy (Academic Press, 1982; 2nd Edition, A K Peters 2001), which in turn attributes it to

• E. Pericoloso Sporgersi.

• Winkler mentions that the phrase is found also on Italian railroad trains, warning passengers not to lean out the window.

The Problem

Alice and Bob alternately choose numbers from among 1, 2, .. ., 9, without replacement. The first to obtain 3 numbers which sum to 15 wins. Does Alice (the first to play) have a winning strategy?

Spoiler

Think magic squares and tic-tac-toe.

Solution

The quick way to solve the puzzle is to imagine that Alice and Bob are playing on the following magic square:

8 1 6
3 5 7
4 9 2

Since it is exactly the rows, columns, and main diagonals which sum to 15, they are playing Tic-Tac-Toe! Therefore neither player has a winning strategy.

Food for Thought

With the same overall game template, what about other sets of numbers, and other targets?