# Finding digits that sum to 15

Rand and Rubio are playing a game in which they each take turns to pick a digit between 1 and 9, without replacement (i.e. all digits chosen are distinct). If one of them manages to get three digits which sum to 15, then he wins. If neither player achieves this, they both lose.

Rand goes first. Does he have a winning strategy? If so, what is it? If not, prove it.

Source: Peter Winkler's Mathematical Puzzles: A Connoisseur's Collection.

• Yes, he said "you take 2, 6 and 7, and I'll take 1, 5, and 9. No one said first to 15 wins so we'll both win this way" – Sandy Chapman Jan 18 '17 at 2:59
• Of course Rand wins - he wouldn't be stupid enough to play against Rubio if he didn't already have a winning strategy worked out! – YowE3K Jan 18 '17 at 3:40
• Does the game end as soon as one of the players get 15? Apparently it doesn't, but still. – Nautilus Jan 19 '17 at 19:25

No.

Explanation:

This game is equivalent to playing noughts and crosses on a 3 x 3 magic square. With perfect play, noughts and crosses is a draw, so the first player does not have a winning strategy.

Proof:

Exactly eight triplets of different digits between 1 and 9 sum to 15:

1 5 9
1 6 8
2 4 9
2 5 8
2 6 7
3 4 8
3 5 7
4 5 6

And...

Exactly eight lines of three numbers - horizontal, vertical, or diagonal - appear on any 3 x 3 magic square. No two are the same.

Therefore...

If the magic square contains all of the digits between 1 and 9, each line contains a different one of the above triplets and all triplets appear. This proves equivalence.

The solution:

No.

The reason:

They're really playing tic-tac-toe on this grid! (All possible combinations with a sum of 15 are contained in it.)
492
357
816

• Are "not losing" and "winning" the same thing? There are plenty of games where neither wins; is it required that Rand have a strategy that can always beat me, or does never losing to me suffice? – Rubio Jan 18 '17 at 1:12
• @Rubio Winning means actually winning. Edited to clarify. – Rand al'Thor Jan 18 '17 at 1:17
• Hmmm... If this is the case, the only way to win is not to play.... – user30599 Jan 18 '17 at 8:04
• I'm not sure if this answer came before @TheEarth's one, but this one is way simpler to understand. Nice! – Xenocacia Jan 18 '17 at 9:34
• @Xenocacia This answer was first, but both are good answers and deserve upvotes. – Rand al'Thor Jan 18 '17 at 12:18