[Guest]

Here's a poker puzzle. It helps if you have a basic understanding of Hold 'Em.

A dinner host has invited N guests to enjoy a meal. At the end of the meal, he will offer each guest a two card Hold 'Em hand. The winner of the hand will receive a nice door prize. He then deals a rigged three card flop and fourth card in the middle of the table (by rigged, I mean he has set the cards ahead of time). If you know Hold 'Em, you know there is one more card to come. The host does not want anyone to share the door prize, so he must make sure that everyone at the table can still win the entire prize (no ties). But he also wants to make sure that there is at least one card left in the deck that will give each person a winner (the remaining cards are randomly shuffled to ensure a fair outcome).

1. Prove the maximum possible value for N. (remember, everyone is drawing to win the entire prize and there is at least one card left that will give them the win). Hint-- N is greater than the normal 9 or 10 players in a typical Hold 'Em game.

2. Solve for N (there are a few solutions). The solution must include the hands dealt to each person, the four cards in the middle, and the remaining cards left in the deck, and which hand will win with which card/s.

This scenario actually came up at a dinner party I was invited to this summer. Ten of us were "dealt" in and with one more card to come, 9 of us were drawing to a win. We thought the host did a nice job with this little game, but wondered what the max N would be. It turns out there is a max N and there is a solution. Enjoy.

[Guest]

it's not possible for N to be greater than 16 for a standard 52 card deck. I've found 11 hands which can each have a unique winning river card given a flop-turn combination, but it seems to get much harder from this point.

[Guest]

for 12 hands (cards designated as rank-suit:

flop/turn: T-H, 9-H, 7-H, 7-S

hand 1: 5-C, 8-C (6-C or 6-D gives winning straight)

hand 2: 7-C, 3-D (3-S, 3-H, or 3-C gives winning full house)

hand 3: 7-D, 4-D (4-C or 4-H gives winning full house)

hand 4: 9-D, 2-D (9-S or 9-C gives winning full house)

hand 5: T-D, 2-S (T-S or T-C gives winning full house)

hand 6: Q-C, J-C (8-S or 8-D gives winning straight)

hand 7: K-D, Q-D (J-S gives winning straight)

hand 8: 6-H, 4-S (8-H gives winning straight flush)

hand 9: K-C, J-D (Q-S gives winning straight)

hand 10: 5-H, 5-D (5-S gives winning full house)

hand 11: A-H, 2-C (2-H, J-H, or Q-H gives winning flush)

hand 12: K-H, 6-S (K-S gives winning 2 pair)

cards remaining:

2-H

3-S, 3-H, 3-C

4-C, 4-H

5-S

6-C, 6-D

8-S, 8-D, 8-H

9-S, 9-C

T-S, T-C

J-S, J-H

Q-S, Q-H

K-S

A-C, A-D, A-S (having all of these Aces in the deck makes me believe there may be a 13th hand possible if things are slightly different..)

Edited November 18, 2009 by ogden_tbsa

[Guest]

for 14 hands (cards designated as rank-suit:

flop/turn: T-H, 9-H, 7-H, 7-S

hand 1: 5-C, 8-C (6-C or 6-D gives winning straight)

hand 2: 7-C, 3-D (3-H gives winning full house)

hand 3: 7-D, 4-D (4-C or 4-H gives winning full house)

hand 4: 9-D, 2-D (9-S or 9-C gives winning full house)

hand 5: T-D, 2-S (T-S or T-C gives winning full house)

hand 6: Q-C, J-C (8-S or 8-D gives winning straight)

hand 7: K-D, Q-D (J-S gives winning straight)

hand 8: 6-H, 4-S (8-H gives winning straight flush)

hand 9: K-C, J-D (Q-S gives winning straight)

hand 10: 5-H, 5-D (5-S gives winning full house)

hand 11: A-H, 2-C (2-H, J-H, or Q-H gives winning flush, A-C gives 2 pair tie with hand 13, hand 14)

hand 12: K-H, 6-S (K-S gives winning 2 pair)

hand 13: A-S, 3-C (A-C gives 2 pair tie with hand 11, hand 14)

hand 14: A-D, 3-S (A-C gives 2 pair tie with hand 11, hand 13)

cards remaining:

2-H

3-H

4-C, 4-H

5-S

6-C, 6-D

8-S, 8-D, 8-H

9-S, 9-C

T-S, T-C

J-S, J-H

Q-S, Q-H

K-S

A-C

[Guest]

Your answer for the max N is correct and I assure you there is a solution for the max. Each player will have exactly 1 card that will give them the whole prize.

for 14 hands (cards designated as rank-suit:

flop/turn: T-H, 9-H, 7-H, 7-S

hand 1: 5-C, 8-C (6-C or 6-D gives winning straight)

hand 2: 7-C, 3-D (3-H gives winning full house)

hand 3: 7-D, 4-D (4-C or 4-H gives winning full house)

hand 4: 9-D, 2-D (9-S or 9-C gives winning full house)

hand 5: T-D, 2-S (T-S or T-C gives winning full house)

hand 6: Q-C, J-C (8-S or 8-D gives winning straight)

hand 7: K-D, Q-D (J-S gives winning straight)

hand 8: 6-H, 4-S (8-H gives winning straight flush)

hand 9: K-C, J-D (Q-S gives winning straight)

hand 10: 5-H, 5-D (5-S gives winning full house)

hand 11: A-H, 2-C (2-H, J-H, or Q-H gives winning flush, A-C gives 2 pair tie with hand 13, hand 14)

hand 12: K-H, 6-S (K-S gives winning 2 pair)

hand 13: A-S, 3-C (A-C gives 2 pair tie with hand 11, hand 14)

hand 14: A-D, 3-S (A-C gives 2 pair tie with hand 11, hand 13)

cards remaining:

2-H

3-H

4-C, 4-H

5-S

6-C, 6-D

8-S, 8-D, 8-H

9-S, 9-C

T-S, T-C

J-S, J-H

Q-S, Q-H

K-S

A-C