# Find the winning strategy for this card game

Nine cards numbered 1 through 9 are kept facing up on a table. Alice and Bob are playing a game wherein they pick up cards one at a time, alternatively. The first person to have three cards with sum 15 wins the game.

Further, it is upto Bob to choose who picks the first card. Can Bob ensure a win?

Note that

(7, 8) is not a winning hand, because two cards have a sum 15
(1,2,3,9) is not a winning hand because four cards have sum 15
(2, 6, 3, 7) is a winning hand because three cards(2, 6, 7) have sum 15

• I swear I've seen this somewhere on the site before.
– user88
Commented Dec 18, 2015 at 8:12
• (or it could just have been my post on PPCG of the same puzzle.)
– user88
Commented Dec 18, 2015 at 8:13

## 1 Answer

This game is actually...

Tic-Tac-Toe (or Noughts and Crosses).
4 9 2
3 5 7
8 1 6
If the numbers are arranged in this 3x3 magic square, all rows, columns, and diagonals (and ONLY those) are sets of 3 that sum to 15.

Therefore,

Bob cannot ensure a win.

• That was quick. Commented Dec 18, 2015 at 5:08
• @Manal: I've been told that before on many of my answers. ;)
– Deusovi
Commented Dec 18, 2015 at 5:08
• That's a "weird" logical jump... How did you determine what the answer "actually" is? Or is this one of those "I know the trick to that..." that you heard somewhere else? Commented Dec 18, 2015 at 13:49
• @WernerCD: When I say "actually", I mean that it's isomorphic - put simply, the games have the same structure. The solution to the more common game in the spoiler block applies to the card game. You could prove the solution using the same tactic but in terms of the cards (without any reference to the simpler game), but it would be more complicated.
– Deusovi
Commented Dec 18, 2015 at 13:53
• I'd like to see another solution for this game in order to be able to use it to show that there is no winning strategy for tic-tac-toe!
– JiK
Commented Dec 18, 2015 at 13:57