[Ganchrow]

This is probably the toughest puzzle yet so I'll just let the last puzzle keep running a bit longer. Respond to this thread if the puzzle wording is at all unclear.

Face-Up Poker

Originally Posted by Rules of Face Up Poker:

The Game
Face-Up Poker is played with two players and a standard 52-card deck. Cards are not dealt at random as in standard poker, instead each player may select any cards he wants from a face-up deck. Each player may always see what the other has drawn. Hands are valued as in standard draw poker.

The Play
Play begins with the first player selecting any 5 cards of his choosing from the deck. The second player does the same with the remaining cards. The first player then discards any number of cards from his hand to the discard pile (just as in draw poker) and then selects that same number of new cards from the deck. The second player does the same.

The Winner
The player with the highest poker hand wins. If both players have the same hand the game is a draw.

Ricky and Steve are playing one another in a game of Face Up poker. Ricky is first to act and Steve second. For his starting hand Ricky has 2,598,960 different choices, only some of which would positively secure him the best possible outcome.

So:

Let X = The number of possible starting hands that Ricky could select in order to secure himself the best possible outcome.

Let Y = The highest 5-card card total of all the possible starting hands that Ricky could select in order to secure himself the best possible outcome. (Ignore suits and count Jacks as 11, Queens as 12, Kings as 13, and Aces as 1. For example, 2C 7D 8S KC AH would be a card total of 2+7+8+13+1=31.)

X * Y = ?

[DrSlamm]

Selecting a royal flush seems like a surefire way to tie

[Ganchrow]

Originally Posted by DrSlamm

Selecting a royal flush seems like a surefire way to tie

Maybe, maybe not.

But even if so, would that represent the "best possible outcome"?

[DrSlamm]

if it is not an automatic TIE i dont understand the game

[Ganchrow]

Originally Posted by DrSlamm

if it is not an automatic TIE i dont understand the game

I think you understand.

If a player held a royal flush at the end of the hand, he could do no worse than tie.

And incidentally, while the puzzle is tricky it is not a "trick question". There is a very logical solution.

[Ganchrow]

Solved by Sean at Sat Jul 14 22:33:01 2006.

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[Ganchrow]

Solved by dvorak51 on Sat Jul 15 at 05:16:22.

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[dvorak51]

I got it too

Of course, I kinda cheated...as this was presented to me in a different form before :P

[Ganchrow]

Solved by DrSlamm on Sat Jul 15 at 06:41:49.

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[DrSlamm]

this puzzle bothered me enough that i could not fall asleep until i had solved it.. it came to me in a stroke of genius

[Ganchrow]

Solved by darkghost on Sat Jul 15 at 08:01:07.

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[darkghost]

Nice little poker puzzle. I really appreciated this one. The poker forum would like this one as well.

[Ganchrow]

Solved by MrX on Sat Jul 15 at 12:08:31.

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[Ganchrow]

Answer: In order to guarantee victory, Ricky would need to select a hand containing four 10s*. This is the only hand which would guarantee Ricky a victory. Hence, Ricky has 48 unique choices for his starting hand. The highest 5-card card total would then be 53 (4 10s = 40, 1 King = 13). So the answer is 48 x 53 = 2544.

*By selecting four 10s, Ricky makes it impossible for Steve to select a royal flush. The best hand Steve could select without any 10s in the deck would be a 9-high straight flush, which would still allow Ricky to select the royal flush after discarding. If Steve instead selected a 4 of a kind higher than 4 10s, then Ricky would just select a straight flush to the card of one lower suit. (For example if Steve had selected 4 aces for his starting hand, Ricky would just choose a king-high straight flush. Without any tens, Steve would be unable to beat Ricky.) END