Pick A Card, Any Card

Logic Level 3

You take these nine cards out of a standard deck (ace through 9 of hearts), put them all face up on a table and play the following game against another player:

Both players take turns choosing a card. The first player to have three cards that add up to 15 wins. The ace counts as one.

If both players play optimally, which player has a winning strategy?

The one that goes first The one that goes second Neither player

1 solution

Geoff Pilling
Jun 15, 2016

Relevant wiki: Combinatorial Games - Winning Positions

The cool part of this problem is when you realize you are just playing a regular game of Tic Tac Toe. Imagine if the cards were laid out as a magic square:

and the winner is the first to pick 3 cards that were in a row. All the combinations that add up to 15 are included in the magic square, which can be counted as follows:

All the combinations which include 1: 1-5-9, 1-6-8
All the combinations which include 2: 2-6-7, 2-5-8, 2-4-9
All the combinations which include 3: 3-5-7, 3-4-8
All the combinations which include 4 (but nothing lower): 4-5-6

And, as is well known for that game $\boxed{\mbox{neither player}}$ can guarantee a win.

One of the coolest solutions I've seen so far on Brilliant. Upvote!

Manuel Kahayon - 4 years, 12 months ago

Thanks, Manuel! :)

Geoff Pilling - 4 years, 12 months ago

Nice Solution! (+1)

Samara Simha Reddy - 5 years ago

Hey thanks, Samara!

Geoff Pilling - 4 years, 12 months ago

$\ddot\smile$

Samara Simha Reddy - 4 years, 12 months ago

For completeness, you have to justify the first line. IE how do we know that every sum to 15 is represented by a line of the grid?

Calvin Lin Staff - 4 years, 12 months ago

This is a wonderful solution.

A Former Brilliant Member - 5 years ago

Thanks, Vighnesh!

Geoff Pilling - 4 years, 12 months ago

Nice solution, very clever!

ozanoz44 UYSAL - 4 years, 12 months ago

@Ozanoz44 Uysal – Thanks! 😉

Geoff Pilling - 4 years, 12 months ago

really a good solution (+1)!! :))

Satyabrata Dash - 4 years, 12 months ago

Hey thanks, Satyabrata! :)

Geoff Pilling - 4 years, 12 months ago

I disagree....only you will see these cards/numbers laid out like a tic tac toe because you created it or wanted the cards/numbers laid out so...looks like you want it to come across as a cryptography or some sorts...but if the cards laid out sequentially like in the picture or randomly.. then the first player is likely to win..

El Sol - 4 years, 10 months ago

Because both players are playing optimally, something as trivial as card position will not affect them. I agree in the real world where players do not play optimally mistakes will be made, just as not every tic-tac-toe game ends in a tie, but that's not what this question is asking.

Anthony Holm - 4 years, 9 months ago

I agree with @El Sol . However you place the cards this is still essentially a tic-tac-toe game.

Geoff Pilling - 4 years, 9 months ago

My belief was that in tic tac toe as long as the first person picked the middle (5) it was garunteed a win.

George Qiao - 4 years, 9 months ago

Actually the first person definitely has the advantage, no doubt about that, but the second person can always force a draw (or a "cats game").

Geoff Pilling - 4 years, 9 months ago

Wished there was a genius upvote on brilliant. This would have been the first solution to get it.

Muralidhar Rao - 9 months, 1 week ago

There are 12 partitions of 15 into 3 distinct integers (=cards). When the first player chooses their first card, the second player can choose one of the integers in the first players sequence either on the first or the second move. By doing this, the second player is choosing a sequence. The first player, seeing they will not win, can choose one of the distinct numbers in the seconds players sequence, so that none of them will be able to complete the sequence of three distinct cards adding up to 15.

Gediminas Sadzius - 3 weeks, 1 day ago

Pick A Card, Any Card

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