# Fast 2014-09-26: Hold ’em logic puzzle #1

#### “Added Fast” purpose:Allow Mike Caro to post spontaneous thoughts, tips, and information.

• Includes Mike’s notes to himself.
• Titles begin with “Fast,” plus date.
• If expanded later, link is at bottom.

Also see:  → Why a Poker1 “Fast” category?  |  → All Poker1 “Fast” entries

This “Added fast” post will probably be elevated soon and become a full Poker1 entry. I think the logic is intact, but I’m not quite confident enough about the phrasing and explanations yet to fully commit to it. That’s the purpose of these fast posts. (In fact, I’ll probably continue to fine-tune the wording.)

Important extra note: The logic originally had a glaring ommission, which required the addition of the clause followed the comma on “Statement of fact 3” below. This was first pointed out by Richard Jenkin on my Facebook page. Please report any other glitches. I’ll also be reviewing this entry carefully myself to make sure it’s 100% correct before committing to it.

This kind of poker puzzle won’t be everyone’s “cup of tea,” so I don’t expect the higher number of visits that other posts enjoy, but if enough people like this, I’ll make it a series.

This puzzle isn’t especially clever. That isn’t the purpose here. There may be a few brain-teaser type clues in some of the puzzles in this series, but the main goal is to help you make conclusions from clues, not to amaze.

Yes, the answer is provided at the bottom, but try to solve it before looking.

Hold ’em puzzle #1

There are 10 clues. All of them may be relevant or only some of them. There is no tricky wording and each statement is true. The puzzle can be solved using the information provided.

Your mission: From the statements below, figure out what cards Amy holds, what the board is, and what five cards are in the winning hand.

Statement of fact 1

Amy knew that if she made her flush on the final river card, she would win the whole pot, unless an opponent made a straight flush.

Note: The above statement was amended 2014-12-22 in accordance with a logical flaw pointed out by R.B. in the comments section below. Unless a link to a full normal entry status exists, this post remains incompletely vetted, as explained in the first paragraph.

Statement of fact 2

Amy missed her flush on the river.

Statement of fact 3

Amy knew, after seeing the river card, that she couldn’t lose to any player who didn’t have a flush of the same suit she missed.

Statement of fact 4

No player held an ace before the flop, although one of Amy’s cards was higher than any non-ace rank on the final board.

Statement of fact 5

The queen of spades appeared on the flop, and it paired only one player.

Statement of fact 6

Amy didn’t hold a spade, but did hold a diamond.

Statement of fact 7

The final river card was a diamond.

Statement of fact 8

Amy held a diamond ranking higher than any diamond on the board, but lower than her other card.

Statement of fact 9

The lowest ranks on the final board were 3 and 8 and they existed prior to the river. In fact, both the highest and lowest rank appeared on the flop.

Statement of fact 10

Bob accidentally exposed the king of hearts and queen of clubs to everyone on the flop.

From these 10 clues:

(1) What were the exact cards (rank and suit) that Amy held?

(2) What five cards were on the board (rank and suit, flop in any order, exact order for turn and river)?

(3) What five cards (rank and suit) did she use that couldn’t be beaten?

Scroll down for the answers and the logic. But, if you haven’t solved it yet, you might want to give it some more thought first.

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

↓ ↓ ↓

Hold ’em Puzzle #1 answers and logic

There is even more that you could divine from the statements. And you could make conclusions by reasoning in different ways and different orders. Below is just one way to think about the puzzle and to solve it.

From statement 1: Amy didn’t have a flush, but had four of one suit before the river card. If she were to make the flush, her hand would be unbeatable.

From statement 2: Amy missed her flush try, so if she won, it was with something else. We can rule out the possibility of her holding suited cards with three members of that suit on board and the possibility of her holding one card of a suit that had four members on board. Either of those circumstances would result in her having made a flush — which she didn’t.

From statement 3: Unless another player had her final hand beat with a flush prior to river card, Amy won or tied and the river card assured that she must. So, either Amy already had the hand and the river card meant that now nobody without a flush could beat it or the river card helped her in a way that meant she couldn’t be beat if she wasn’t already. Which? We don’t know yet.

From statement 4: Amy didn’t hold an ace, but held at least one card higher than the second-ranking card on board. This tells us that if we can figure out any rank on board, Amy has at least one private card outranking it.

From statement 5: That was fast! Already we know a queen was on the board, so Amy must have held a king (because an ace has already been excluded).

From statement 6: Amy didn’t hold a spade, so she couldn’t have been trying for a spade flush that would guarantee a win. She held a diamond, though. It might have been the king that we know she holds or her other card.

From statement 7: The final board card was a diamond, but Amy didn’t make a flush, so there could not have been two diamonds on board before the river, assuming she held two diamonds privately, or three diamonds on board before the river, assuming she held one diamond privately. Her flush try must have been hearts or clubs, since statement 6 told us bluntly that she didn’t hold a spade.

From statement 8: So, we already know Amy didn’t have two diamonds or she would have made a flush, because she was trying for an unbeatable one. So, she held one diamond, but that couldn’t have been the suit she was trying to make the flush in or she would have connected. Therefore, it’s the other suit among her two private cards that she was trying to make the flush in. But the diamond she did hold was higher than the river card, since that was a diamond.

From statement 9: The lowest ranks on the board were 3 and 8, and they were there before the final card. So, the final diamond must have been at least a nine.

From statement 10: Bob held the only heart. So, that leaves clubs as the only suit in which Amy could have been trying for a flush. Since she had two different suits, that means means exactly three clubs must have been present on the board before the river. Since Amy didn’t hold an ace, but had a chance to make an unbeatable club flush with any club landing on the river, that means she must have the king of clubs. We already know there’s a queen of spades on the board. So that means the other three clubs must be ace, plus the two lowest ranks, 3 and 8. Since she made a hand that couldn’t be beaten on the river, it must be a straight. Had there been a pair on the board, then she couldn’t have been sure that making her flush would win. We know her diamond didn’t rank as high as her other card, so that excludes her having a private pair. You might reason that she could hold either K♣-J♦ or K♣-10♦ and have made an ace-high straight. But that’s wrong, because K-10 is excluded. Why? It’s because her diamond is higher than any on the board, so that means J♦ vs. a 10♦ river card. The river diamond only needed to be higher than an 8, but nine doesn’t work, because it doesn’t afford Amy the chance to make the unbeatable ace-high straight. Finally, clue 10 tells us that Bob (another player) exposed the queen of clubs earlier. That’s important, because otherwise, Amy would not know that her flush would win if a queen of clubs hit on the river. She could then have been beaten by a full house or four queens. So, statement 10 closes the remaining gaps.

So:

(1) Amy’s hand is K♣ J♦

(2) The board consisted of A♣ Q♠ 3♣ 8♣ 10♦

(3) She won or tied with this ace-high straight: A♣ K♣ Q♠ J♦ 10♦

— MC | Follow-up link: → None

Also see:  → Why a Poker1 “Fast” category?  |  → All Poker1 “Fast” entries

### Mike Caro

Twitter: http://www.twitter.com/mikecaro FaceBook: http://www.facebook.com/caro.mike Known as the "Mad Genius of Poker," Mike Caro is generally regarded as today's foremost authority of poker strategy, psychology, and statistics. He is founder of Mike Caro University of Poker, Gaming, and Life Strategy (MCU). See full biography at Poker1.com.

## 6 thoughts on “Fast 2014-09-26: Hold ’em logic puzzle #1”

1. R.B. says:

Sorry to nitpick. Statement #1 is incorrect because it is possible for someone other than Amy to make a straight flush if the river card is the 2, 4, or 5 of clubs. In those cases she would have made her flush, but would not have won the whole pot.

1. Hi, R.B. —

You are absolutely correct. I have amended the first statement of fact to add “, unless an opponent made a straight flush” before the period.

This is why I stated in the opening paragraph that I wasn’t confident enough to elevate the logic problem to regular entry status. It remains a hastily created prototype for a possible future series, in which the answers and logic will be fully vetted.

I greatly appreciate your input and have credited you with the correction.

Straight Flushes,
Mike Caro

1. R.B. says:

Thank you for your response, and the work you do as a teacher. I am a big fan of yours and glad I found this site. I enjoy poker and logic problems, and would love to see these puzzles as a series.

2. Here’s a puzzle for you Mike Caro – I’m fast posting so hope this works…

A table is about to be broken. The tournament director separates out the thirteen clubs and places 10 of them on the players’ baseball hats so that only the player allocated the card can’t see it. He places the remaining three cards on the table, two are face down, one is the jack of clubs.

The player with the highest rank card is to move to table one, the player with the next highest ranked card is to move to table two, etc.

The players can move as soon as they know the rank of their card.

The players can’t communicate with one another.

The tournament director did not choose the cards randomly.

After all the players take their new seat the two face down cards are revealed as?

3. P.S. The explanation, reasoning and deductions I thought were right on. In Re: to you not being confident about phrasing and explanations.

4. Loved the puzzle! What a great way to practice deductive reasoning, which is something I had no idea how to “practice” before. My opinion might not be much, but I would say yes please on more!