### Brain Teasers

# Five Cards

I was playing a game of five card draw poker with a bunch of logicians. By the time we had finished bidding and were just about to reveal our cards, I was pretty confident I would win of the four of us remaining. I had three nines, some face card (I can't remember what suit or even whether it was a jack, queen, or king) and a four. (Or was it a five? I can't remember.)

I was even more sure when two of my opponents laid down their cards. One had a pair of fours and a pair of sevens, the other had a pair of twos and a pair of eights. My third opponent, however, laid down his five cards face down in a row. He said, "I have a straight, and the cards are, from lowest to highest: a ten, a jack, a queen, a king, and an ace. I have at least one card of each of the four suits: clubs, spades, hearts, and diamonds. I am fairly certain that this is the winning hand, but I'm feeling generous today, and I will give a third of the pot to whoever can determine which suit I have two cards of.

Now I know you can't figure it out without some clues. Here they are:

1. The king is next to at least one diamond.

2. The queen is next to exactly one heart.

3. The jack is next to at least one spade, but is not next to any hearts.

4. The ten is next to at least one club.

5. The ace does not border any black cards, nor does it border any diamonds.

6. My two cards of the same suit are not next to each other.

7. Of the ten possible pairings of cards, only one pair, when removed, leaves three cards in ascending order from left to right.

8. My ace is not the card on the far left."

There was a minute's silence. One of the other logicians said, "I give up! There's no way to figure that out!"

The other agreed. But I didn't. I had just figured out which suit he had two of.

Which suit is it?

I was even more sure when two of my opponents laid down their cards. One had a pair of fours and a pair of sevens, the other had a pair of twos and a pair of eights. My third opponent, however, laid down his five cards face down in a row. He said, "I have a straight, and the cards are, from lowest to highest: a ten, a jack, a queen, a king, and an ace. I have at least one card of each of the four suits: clubs, spades, hearts, and diamonds. I am fairly certain that this is the winning hand, but I'm feeling generous today, and I will give a third of the pot to whoever can determine which suit I have two cards of.

Now I know you can't figure it out without some clues. Here they are:

1. The king is next to at least one diamond.

2. The queen is next to exactly one heart.

3. The jack is next to at least one spade, but is not next to any hearts.

4. The ten is next to at least one club.

5. The ace does not border any black cards, nor does it border any diamonds.

6. My two cards of the same suit are not next to each other.

7. Of the ten possible pairings of cards, only one pair, when removed, leaves three cards in ascending order from left to right.

8. My ace is not the card on the far left."

There was a minute's silence. One of the other logicians said, "I give up! There's no way to figure that out!"

The other agreed. But I didn't. I had just figured out which suit he had two of.

Which suit is it?

### Hint

The first step is to determine where the ace goes.Also, consider this: How could I have figured it out when two expert logicians couldn't?

### Answer

The two cards on the ends and the card in the middle border a combined number of two suits. So at least one pair of them should be listed as bordering a card of the same suit in clues 1-5. Of the five, only the ace and the queen meet this criterion, so they must be two cards apart. One of them must be in the center, but it can't be the ace (the two cards on the ends would only border hearts - and one of them has to be bordering something else from clues 1-4), so it must be the queen. The ace is not on the left end from clue 8, so it must be on the right end, and the card next to it must be a heart. So far we have this(X represents unknown):X X Q X A

X X X H X

At least one of the two cards to the left of the queen is lower than the queen (two of the remaining cards are lower, and only one can be right of the queen), so the triplet that is in ascending order is the queen, the ace, and the single card to the left of the queen that is lower than the queen. That means the king is left of the queen, and is also left of both the jack and the ten (otherwise, jack king ace or ten king ace would be a second triplet), so it must be on the far left. The jack must be left of the ten (or else ten jack ace would be a second triplet), so it must be second from the left, and the ten must be second from the right. The only card next to the king is the jack of diamonds (it's a diamond from clue 1). So far we have(T is ten):

K J Q T A

X D X H X

Now we seem to be stuck. We don't have any more clues that can be used. So how did I figure it out?

I held a face card (I said so myself in the intro). I must have known it cannot be that card and deduced the correct answer from there.

But which card did I have? We already know the suit of the jack, so my holding a jack would not help. Could I have held a queen? No, because I know the queen to either be a spade or club (clue 6), so one of the cards next to the queen would have its clue satisfied (see clues 3 and 4), and I could not determine the suit of the card on the other side of it.

Therefore, I must have held a king. But which king? I know the king is not a diamond (clue 6) or a heart (clue 3). If I held the king of clubs, then the king next to the jack would have satisfied the jack's clue, and I could not have determine the suit of either card next to the ten.

Therefore, I held the king of spades. The king of clubs must have been on the far left (only suit available), so the only card next to the jack that could be a spade is the queen, so the only card next to the ten that could be a club is the ace. In summary:

K J Q T A

C D S H C

Therefore there are two CLUBS.

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## Comments

My first teaser!

Wow! that was well thought out and well done.

Very good.

This reminds me of the Potions "Task" in Harry Potter and the Sorcerer's/Philosopher's Stone that Hermione solved on the way to get the Stone.

In what way?

wow, good job Poker. Rarely is someone's first teaser such a good one!

Thanks, Mog! It's one I created some time ago. Glad you like it!

Others I have made, on the other hand, aren't quite as good.

Others I have made, on the other hand, aren't quite as good.

Well done, especially in determining the suits for the king, queen, and ace. I do have one problem with the teaser. The second statement about the queen being next to exactly one heart led me to believe that the queen had to be on either end, especially since other statements used the phrase 'next to at least'.

I said next to exactly one heart. I didn't say about any of the other suits.

This teaser did not explain itself enough. Nowhere in the teaser does it say that the cards weren't in order, but it also doesn't say they were. So, that can't be faulted. The explaination made my brain hurt! The first part about how the queen and ace ended up where they were made no sense. Another person did the teaser on their own and came up with the same answer I did, so it isn't just me.

Sorry! Didn't mean to put my comment on here three times.

Let me explain it to you again. The card on the left end borders a card of one suit - call it suit A. The card on the right end borders another suit - call it suit B. Suits A and B could be the same suit, but that's beside the point. The point is, the card in the middle borders a card of suit A and a card of suit B, since there are five cards. The card on the left will be listed as bordering suit A, the card on the right will be listed as bordering suit B, and the card in the center will be listed as bordering either suit A or suit B. So one of suits A and B will be listed twice. The only suit listed as bordering two cards is hearts - and it borders the ace and the queen. The ace can't be on the left side, nor can it be in the center (the card on the end that is not the queen borders something other than hearts), so it must be on the right, with the queen in the center. Which explanation do you prefer - this one or the one in the answer?

SECOND ANSWER:

JKQTA

HSDHC

BECAUSE THERE IS A SECOND POSIBLE ORDER OF CARDS WHICH MAKE THE REST OF YOUR REASONING INVALID BECAUSE THERE ARE TWO POSIBLE ORDERS OF CARDS BUT IF YOU CONTINUE WITH THE NEW ORDER YOU WILL COME TO THE CONCLUSION THAT YOU HAVE A JACK OF DIAMOND IN YOUR HAND WHICH WIL LEAD YOU TO THE CONCLUSION THAT HEARTS IS THE ANSWER

JKQTA

HSDHC

BECAUSE THERE IS A SECOND POSIBLE ORDER OF CARDS WHICH MAKE THE REST OF YOUR REASONING INVALID BECAUSE THERE ARE TWO POSIBLE ORDERS OF CARDS BUT IF YOU CONTINUE WITH THE NEW ORDER YOU WILL COME TO THE CONCLUSION THAT YOU HAVE A JACK OF DIAMOND IN YOUR HAND WHICH WIL LEAD YOU TO THE CONCLUSION THAT HEARTS IS THE ANSWER

woah

I couldn't answer it if you would pay me. It was cool though

Line 7 really threw me off. I never quite understood what that meant.

Apr 01, 2005

that teaser was so easy.why does everyone think its hard.

Sorry, but your "second answer" doesn't meet the conditions of clue 7.

I came up with the answer

TJQKA

DCSHD

Is this one wrong?

the ONE and ONLY

Stan D Man

TJQKA

DCSHD

Is this one wrong?

the ONE and ONLY

Stan D Man

I'm lost,

I guess this is why the only card games I play is solitare and battle and BS.

I guess this is why the only card games I play is solitare and battle and BS.

Once again, clue 7 is not met.

wow poker, that was really hard, thats the only logic problem that I tried that I couldn't figure out.

excellent i didn't get it in my self imposed 2 minutes, so when i saw how you worked the clues i'm glad i gave up,,,very well done

wow,,i didn't notice how old this one was ,,i was just searching the ones rated most difficult

A good one, and it's not easy; it took me 15 minutes to solve it.

Keep up the good work

Keep up the good work

Your Clue #7 is confusing. The way you make it sound, there can be ONLY ONE sequence of cards that will satisfy Clue #7, when, in actuality, there are other sequences that will fit this clue.

Dec 21, 2005

Actually hawkinsunc, you are incorrect. There is only one sequence that satisfys Clue #7. Any other sequence than the correct one is either not able to be put in an Ascending Order from Left to Right (AOL2R) at all, or, can have two different cards removed and still be in AOL2R. E.g.: TJQKA, can have multiple pairs of cards removed and still be in AOL2R...Remove TJ and QKA is still in AOL2R, remove QK instead and TJA is still AOL2R. Conversely, the correct sequence, can only have two specific cards removed and still be in AOL2R. Additionally, as an example, with the following sequence, there is no way to ever get it into AOL2R, no matter which two cards you remove: AQKJT.

that was a very good teaser... yeah I know that I'm late to seeing it but I just joined this site not too long ago... I nearly got it out but what got me was the last part where you had to review the cards that were in your hand and not only the cards in ur opponents hand. VERY GOOD

Aug 20, 2006

yeah clue 7 is not met, and i had a deck to try it out, and came up with something for the first 6, but 7 made it false but it was a good one thou

E-Z But very fun and well thought out

No way to determine the order of the cards, why couldn't they be in order?

easy for me

The way I see it, there are five different possible answers:

K J Q 10 A

S D C H S

S D C H C

S D S H C

C D S H C

S D C H D

I believe all five of those options satisfy the conditions stated in the clue. So he could have two clubs or two spades or two diamonds.

Also, the problem states that with regards to the face card in your hand you "can't remember what suit or even whether it was a jack, queen, or king". This to me means that the identity of the face card is unknown to you when trying to reason which suit appears twice in the hand. So I don't see how you could figure out that it was a King or which suit it is without guessing.

K J Q 10 A

S D C H S

S D C H C

S D S H C

C D S H C

S D C H D

I believe all five of those options satisfy the conditions stated in the clue. So he could have two clubs or two spades or two diamonds.

Also, the problem states that with regards to the face card in your hand you "can't remember what suit or even whether it was a jack, queen, or king". This to me means that the identity of the face card is unknown to you when trying to reason which suit appears twice in the hand. So I don't see how you could figure out that it was a King or which suit it is without guessing.

Great teaser! Took a while to get it.

There are 24 possible orders for the cards to meet clues 7 & 8:

TKAQJ

TAQKJ

TAKJQ

JKATQ

JKTAQ

JKAQT

JKQTA

JKQAT

JAQKT

JAQTK

QKATJ

QKAJT

QKTAJ

QKJTA

QKJAT

QTJAK

QTAJK

QATJK

KTQAJ

KJQTA

KTAJQ

KQTJA

KJQAT

KATJQ

Of these only two can satisy the rest of the conditions:

JKQTA

KJQTA

For JKQTA the pattern must be XSXHX. There are six ways to complete this pattern to satisfy the conditions:

JKQTA

DSDHC 2D

CSDHC 2C

HSDHC 2H

DSCHD 2D

DSCHC 2C

DSCHS 2S

There is no face card that the player can hold that will produce a single answer. (Jack of diamonds allows either clubs or hearts.)

For KJQTA the pattern must be XDXHX. There are five ways to complete this pattern to satisfy the conditions:

KJQTA

CDSHC 2C

SDCHC 2C

SDSHC 2S

SDCHD 2D

SDCHS 2S

Only the king of spades reduces this to a single possibility, which is of course clubs.

There are 24 possible orders for the cards to meet clues 7 & 8:

TKAQJ

TAQKJ

TAKJQ

JKATQ

JKTAQ

JKAQT

JKQTA

JKQAT

JAQKT

JAQTK

QKATJ

QKAJT

QKTAJ

QKJTA

QKJAT

QTJAK

QTAJK

QATJK

KTQAJ

KJQTA

KTAJQ

KQTJA

KJQAT

KATJQ

Of these only two can satisy the rest of the conditions:

JKQTA

KJQTA

For JKQTA the pattern must be XSXHX. There are six ways to complete this pattern to satisfy the conditions:

JKQTA

DSDHC 2D

CSDHC 2C

HSDHC 2H

DSCHD 2D

DSCHC 2C

DSCHS 2S

There is no face card that the player can hold that will produce a single answer. (Jack of diamonds allows either clubs or hearts.)

For KJQTA the pattern must be XDXHX. There are five ways to complete this pattern to satisfy the conditions:

KJQTA

CDSHC 2C

SDCHC 2C

SDSHC 2S

SDCHD 2D

SDCHS 2S

Only the king of spades reduces this to a single possibility, which is of course clubs.

I didn't make it clear in my previous comment that having the king of spades in my hand would also eliminate all of the JKQTA combinations since they all require the king of spades.

Just joined and this is one of my first teasers...

You say the answer is:

KJQTA

CDSHC

How does this satisfy the seventh citeria:

7. Of the ten possible pairings of cards, only one pair, when removed, leaves three cards in ascending order from left to right

After removing the pair of suited cards (the only pairing) it leaves you with:

JQT

How are those three cards in ascending order from left to right ?

ascending order would be TJQ !

The answer I came up with is:

TJQKA

SCDHS

After a quick double-check it seems to satisfy all criteria.

Not to mention most poker players lay down their straights in ASCENDING SEQUENTIAL ORDER !

With that rational taken into consideration I think that is the simplest answer and therefore the best answer. I didnt bother to look for a second answer for that same reason - the simplest answer is the best answer. Well, that and I just didn't want to spend any more time on this...

You say the answer is:

KJQTA

CDSHC

How does this satisfy the seventh citeria:

7. Of the ten possible pairings of cards, only one pair, when removed, leaves three cards in ascending order from left to right

After removing the pair of suited cards (the only pairing) it leaves you with:

JQT

How are those three cards in ascending order from left to right ?

ascending order would be TJQ !

The answer I came up with is:

TJQKA

SCDHS

After a quick double-check it seems to satisfy all criteria.

Not to mention most poker players lay down their straights in ASCENDING SEQUENTIAL ORDER !

With that rational taken into consideration I think that is the simplest answer and therefore the best answer. I didnt bother to look for a second answer for that same reason - the simplest answer is the best answer. Well, that and I just didn't want to spend any more time on this...

G-RAN-IT: Nothing in the seventh criteria restricts the pairing of cards removed to be of the same suit. So there are (as the criteria states) ten possible pairings. The only pair of cards that can be removed from KJQTA to leave ascending cards is KT, leaving JQA. ANY pair of cards removed from your solution leaves three cards in ascending order.

That last comment didn't carify anything...

Your statement doesn't take into consideration that the cards are to be ordered:

The only pair of cards that can be removed from KJQTA to leave ascending cards is KT, leaving JQA.

JQA are not an order. JQK would be an order.

and if ANY TWO cards can be removed why is it referred to as a pair. A pair by definition is:

Two corresponding persons or items, similar in form or function and matched or associated.

how are KT similar or associated ?

they are about as similar as ANY two cards are in the deck, common only in that they are both cards in a deck.

I still stand by my last comment - the best answer is the simplest and I believe I found that. Not to mention I don't believe the original answer is even correct - based partially on the fact that a pair needs to have something in common such as a suit, which I believe was the main factor that we were after - and it just so happens that there are only 10 possibilities of suited pairs.

I could go on but i won't...

Your statement doesn't take into consideration that the cards are to be ordered:

The only pair of cards that can be removed from KJQTA to leave ascending cards is KT, leaving JQA.

JQA are not an order. JQK would be an order.

and if ANY TWO cards can be removed why is it referred to as a pair. A pair by definition is:

Two corresponding persons or items, similar in form or function and matched or associated.

how are KT similar or associated ?

they are about as similar as ANY two cards are in the deck, common only in that they are both cards in a deck.

I still stand by my last comment - the best answer is the simplest and I believe I found that. Not to mention I don't believe the original answer is even correct - based partially on the fact that a pair needs to have something in common such as a suit, which I believe was the main factor that we were after - and it just so happens that there are only 10 possibilities of suited pairs.

I could go on but i won't...

A pair of cards, a pair of ducks, a pair of coins...doesn't seem all that semantically challenging to me to have a pair of cards. If two of the cards had the same rank, then the wording (which could have been better) would be wrong because with cards, a pair means the same rank.

Also, "ascending order" does not say "consecutive ascending order".

You're reading more into the words than is there. Your solution clearly doesn't meet the criteria.

Also, "ascending order" does not say "consecutive ascending order".

You're reading more into the words than is there. Your solution clearly doesn't meet the criteria.

Jul 02, 2010

Poker, you rock!

None of these people have a clue.

Yes, there are 5 possible solutions once you get to the end. BUT Poker said SHE KNOWS the answer. There is only ONE solution in which she could KNOW the answer and that's for her to have the Kspades.

Also, drop it with clue 7. It was fine.

Everyone is saying....

You can have JTKQA....or something. NO you can't. You can remove the T and K and you have JQA...OR you could remove T and Q and you have JKA or you could remove the J and Q and you have TKA.

The only way for it to work is to have KJQTA because you can only remove K and T to get JQA. There is no other order that works.

None of these people have a clue.

Yes, there are 5 possible solutions once you get to the end. BUT Poker said SHE KNOWS the answer. There is only ONE solution in which she could KNOW the answer and that's for her to have the Kspades.

Also, drop it with clue 7. It was fine.

Everyone is saying....

You can have JTKQA....or something. NO you can't. You can remove the T and K and you have JQA...OR you could remove T and Q and you have JKA or you could remove the J and Q and you have TKA.

The only way for it to work is to have KJQTA because you can only remove K and T to get JQA. There is no other order that works.

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