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Me and Varun playing a game of cards (standard deck of 52 cards). In which we distributed 5 cards to each other. Edit: We have seen each other's cards. We have to put the cards in a line one by one alternatively. We don't make a separate line, we put the card in the common line. And we made the winning rule that one who makes a pattern (given below) will win the game.

Pattern:
XX
or
XYX
or
XYYX
and so on, where X and Y are the card name (suites are not distinguished.)

Now when we were beginning the game, we argue on, who's turn will be first. And finally, we came up with I will show the card first. But I paid for this, that I will only win if I make the pattern, and Varun will win if he made the pattern first or Varun will win even when no one makes the pattern and cards finish to both of us.

Now you have to tell me, how best card can I choose from available so that winning chances of mine are maximum?

(Give a proper explanation)

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  • $\begingroup$ How are the cards distributed? Randomly, or do you get to choose? If you do, in which order does the choosing happen, and do you get to see which card the other guy chose immediately? $\endgroup$ – Bass Jan 1 '18 at 18:20
  • $\begingroup$ Cards are distributed from a well-shuffled deck. $\endgroup$ – Siraj Jan 1 '18 at 18:27
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Since you have seen each other’s cards, this is a game of full information, so assuming perfect play from each player, the ”winning chances” are either ”sure win” or ”sure loss”. Probabilities won’t need to be involved.

Case 1: (”trivial win”) you have a pair (or more) of cards the opponent doesn’t have. Strategy: play one, win on your next turn by playing another.

Case 2 (special case): Varun has a pair of cards you have one of, the reverse doesn’t hold, AND Varun doesn’t have a trivial win in the other cards. Strategy: play the cards V doesn’t have (you are guaranteed to have more than V), win when he has to play a card you are holding.

Case 2 (general case): You each have some cards in common with each other. You have more cards that are not shared. V does not have a trivial win on his turn. Strategy: play the non-shared cards, win.

Case 3 (all other card distributions): You’ll lose, no matter what you play.

The final case follows from the setup: if you don’t win, you’ll automatically lose. Since someone has to play the other card before you can win, cases 1 and 2 are the only ways to win, unless V makes a mistake.

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You should play a card that you hold at least two of. For instance if you hold two 10s then you should play a 10. This is because if you have a pair of 10s, or better yet three 10s then Varun has a greatly diminished opportunity to create win pattern #1 (XX) but you have a 100% opportunity to create win pattern #2 (XYX) where it doesn't matter what card Varun plays for (Y) as long as it is not a 10.

Win pattern #3 would not happen if you follow the above rule and strategy. It also seems that as soon as you see each other's cards the game is almost moot...why play? If you are playing the first card and you have a pair of cards that Varun has none of then you win!

Edit: If you do not have a pair: You should first play a card that Varun does not have a match to because he would quickly win. Instead play a card that he cannot match but that you can negate his play until all cards are played and achieve a stalemate.

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  • $\begingroup$ You're close, but not there. $\endgroup$ – Siraj Jan 1 '18 at 18:28

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