# Tricky Logic Question

Discussion in 'Odds & Ends' started by Claudius, May 1, 2011.

1. ### ClaudiusProfessional

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Students X, Y and Z are good at logical reasoning. They already know that the drawer has the following 16
playing cards:

Heart (♥): A♥、Q♥、4♥
Club (♣): K♣、Q♣、5♣、4♣、6♣
Diamond (♦): A♦、5♦

Teacher W now picks one card out of these 16 cards from the drawer without X, Y and Z seeing it.
Then W tells the rank of this card (i.e., A, 2, 3, 4, 5, 6, 7, 8, J, Q, or K) to Y and the suit of this card
(i.e., Heart, Spade, Club, or Diamond) to Z. Then he asks Y and Z whether each of them can infer
what this card is just from the partial information they each have. At this point, X hears the following
conversation between Y and Z:

Y: I don’t know what this card is.
Z: I know that you don’t know what this card is.
Y: Now I know what this card is.
Z: Now I also know what this card is.

After hearing the above conversation and thinking for a while, X also knows what this card is.

Don't bother googling; the answer isn't online.

2. ### RoddickAceHall of Fame

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Five of diamonds.

Z can only know that Y doesn't know what the card is, if the suit is not spades or clubs, since there are card ranks in those suits that have no duplicate.

Y can only know what suit the card is if within hearts or diamonds, there is no duplicate in those 2 suits. Therefore, it cannot be the ace.

Z can only know which card it is if within either hearts or within diamonds, he has only 1 choice. So it must be the 5 of diamonds.

3. ### spacediverHall of Fame

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hmmm... the only way that Y would know the card's identity is if she was told that it was a 2, a 3, a 6, a 7, an 8, a Jack, or a King. This is because there is only one of each of those ranks in the drawer.

Z knows this too.

Therefore, the only way that Z would know that Y doesn't know the identity, is if the card is NOT one of these seven ranks. In other words, Z somehow knows that the card is an Ace, a 4, a 5, or a Queen.

The only way that Z would know that the card is an Ace, a 4, a 5, or a Queen is if he was told that it was a heart, or he was told it was a diamond.

Y knows this.

So Y now knows that the card is either a heart or a queen, and had previously known that the card was an Ace, a four, a five, or a queen.

If she were told it were an Ace, then knowing it was diamonds or hearts wouldn't help her, since there is an ace of diamonds and an ace of hearts.

If she were told it were a queen, she'd know that it was a queen of hearts.

If she were told it were a four, she'd know it was a four of hearts.

If she were told it were a five, she'd know it was a diamond.

Z knows this.

So we, the solvers of this puzzle, now know that the card must be one of these three possibilities:

a queen of hearts
a four of hearts
a five of diamonds

Remember, Z knows which suit the card is.

If Z were told the suit were hearts, he wouldn't be able to distinguish the first of the two possibilities (queen of hearts or four of hearts).

So the only way Z knows the card is if he were told it were a diamond, and now he knows that the card is a five of diamonds.

4. ### spacediverHall of Fame

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lol damn u roddickace.

i was figuring the puzzle out as i wrote my post, so it took me a bit longer

(but i guess u had to figure it out before writing ur post, so u win)

5. ### RoddickAceHall of Fame

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XD I rushed through my explanation to get it in first LOL

6. ### baek57Professional

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It has to be between A Q 5 4 because the first line says "Y: I don’t know what this card is." and those are the only duplicate numbers for him not to know.

Heart (♥): A♥ Q♥ 4♥
Club (♣): Q♣ 5♣ 4♣
Diamond (♦): A♦ 5♦

The second line says "Z: I know that you don’t know what this card is." but it does not say "Z: I don’t know what this card is." So the second line does nothing more than repeat the first line and adds nothing of value.

So one begs to question how Y could possibly jump from "Y: I don’t know what this card is." to "Y: Now I know what this card is." when nothing of any substance was said.

However, if we assume that Y and Z both somehow figured out what the card is from the above information, there is only one possible answer. That is 4♠. This is because it is the only card that can be isolated as the correct answer since all other spades have been eliminated as possible correct answers and at no point in the conversation did Z say he did not know the answer.

7. ### ClaudiusProfessional

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Y saying that he doesn't know what the card is initially indicates that the card has a duplicate. Z already knew that Y didn't know what the card was. This rules out all the spades and clubs, since some of the spades and clubs don't have duplicates. (Remember Z was told only the suit). So that leaves us with the hearts and diamonds.

Now Y could tell which card it was after being told by Z that he knew that Y didn't know which card it was. The possibilities are now narrowed down to the Queen of hearts, 4 of hearts, and 5 of diamonds (since if Y was told that it was an Ace, he would still be uncertain).

Z is able to tell which card it was, right after Y figured it out. This means he wasn't told it was hearts - it has to be diamonds (since if Z was told hearts, he would still be uncertain at this point).

It follows that the card is 5 of diamonds.

8. ### aceXHall of Fame

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Does Z know that Y does not know what the card is before or after the first statement?

9. ### ClaudiusProfessional

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Yes, Z knew before the first statement.

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8♠

10char

11. ### baek57Professional

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It doesn't rule either spades nor clubs. Both Y and Z know all 16 cards before one is drawn. Since Y was only told the number and not the suit and is still unsure what the real card is, we know it can be any one of A Q 5 4, which are present in all suits. You cannot just randomly exclude spades and clubs because there is nothing in your conversation that warrants them to be excluded. The only possible person who is in any position to know what the card is at this point is Z, if he is told it is a spade. If he was told it is anything else neither Y or Z could possibly figure out the true card given the information in your problem. Thus, since you have said that they know the card that was drawn, the only logical conclusion is that the card is 4♠.

12. ### ClaudiusProfessional

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Put yourself in Z's position. If you were told that the suit is diamonds, you can be certain, before having any contact with Y, that Y will not know which card it is, since all diamonds have duplicate ranks. Same goes for hearts (all hearts have duplicate ranks). Now if you were told that the suit was either clubs or spades, you wouldn't be certain that Y doesn't know which card it is, since if he was told that the rank is either J, 8, 2, 7, 3, K, 6, he would know which card it is.

Z knows that Y doesn't know => The suit cannot be spades or clubs

13. ### baek57Professional

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Whether Z knows if Y knows which card it is beforehand is irrelevant because Y tells him that he does not know which card it is. That goes to say that the 2nd statement in the conversation is redundant since Y already told him he didn't know.

Put yourself in Z's position and you were told it was a spade. It goes to follow that Z could conclude based on Y's not knowing the card that the card is 4♠.

14. ### pinky42Rookie

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Perhaps it would have been clearer if the second statement (first by Z) was:
I knew that you couldn't know what this card is.

The way it's phrased is making baek57 think that Z is basing his statement from what Y just said when the intent was show that Z knew all along that Y had no chance of knowing the card from the very start.

That being said, 5 of diamonds is the answer for the reasons given above.

15. ### ClaudiusProfessional

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Okay, revised second statement: I knew that you don't know what this card is.
Is that better?

You can argue all you want, you'll still be wrong. This question has been settled.

16. ### RoddickAceHall of Fame

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Any other ones?

17. ### ProgressoRHall of Fame

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ok so they are good at logical reasoning but what if one of them was trippin'?

Given they are students there is fair chance of that.

I'm going for the King of Clubs?

Is I right??

Is I?

18. ### baek57Professional

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And you can argue all you want but the way you worded it was wrong. Your revision is better for your desired answer, even though it is grammatically incorrect, I understand the change that you were trying to make. Small changes in wording have large impacts on problems like these and in communication in general.

Example: if I change the 3rd and 4th statements with each other using your exact wording, what answer do you get?

Y: I don’t know what this card is.
Z: I know that you don’t know what this card is.
Z: Now I also know what this card is.
Y: Now I know what this card is.

Compare that with changing the 1st and 2nd statements with each other.

Z: I know that you don’t know what this card is.
Y: I don’t know what this card is.
Y: Now I know what this card is.
Z: Now I also know what this card is.

Or Compare that with changing the 2nd statement from present tense to past tense as you did in your revision.

Y: I don’t know what this card is.
Z: I knew that you didn’t know what this card is.
Y: Now I know what this card is.
Z: Now I also know what this card is.

You should not come up with the same answer for all three layouts.

19. ### ClaudiusProfessional

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Not sure what you're trying to show by switching statements. You wouldn't get a definite answer for any of those layouts.

If I say, "I don't know the answer," and you respond by saying, "I know that you don't know the answer," it's rather obvious that you knew that I didn't know before I even spoke.

This puzzle was proposed by world-renowned logician Raymond Smullyan, so if you don't like the wording, take it up with him.

I think it's perfectly fine as it stands.

20. ### baek57Professional

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Actually it's rather obvious that it's not clear given that multiple people are questioning the intent of the second statement.

You make a leap of faith going from "I know that you don't know the answer" to "I knew that you didn't know the answer" based on what you think the author intended. However, the two statements are not inherently connected nor are they equivalent.

If you think it's perfectly fine as is then good for you. However, a well written argument should have no ambiguity as to the author's intent.

And yes, you would get a definite answer with those layouts.

21. ### ClaudiusProfessional

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Well, at least for the first one, there is no answer.

Z couldn't have possibly figured out which card it was after just one statement from Y.

22. ### spacediverHall of Fame

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agreed that the wording should have been less ambiguous. I had to make the assumption that Z already knew that Y didn't know the card, even before Y told Z.

23. ### ProgressoRHall of Fame

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baek57 I think you are barking up the wrong tree, I dont see the statements as ambiguous or redundant.

Each statement makes sense and in the order they are presented, provides the thinker with useful information in turn.

You seem to have several issues so it's hard to pinpoint exactly what your concerns are, but I am with Claudius on this one, it's a perfectly well stated puzzle.

24. ### baek57Professional

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Yep, not ambiguous at all... considering over half the people who posted in this thread questioned the intent of that 2nd statement.

He could if he were told it was a spade.

25. ### GPBProfessional

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Ok so I promise I haven't looked at ANY answers yet. Here's how I'm thinking...

1. Y: I don’t know what this card is.: Y has a 4, 5, Q, or A.
2. Z: I know that you don’t know what this card is.: The suit is heart, since all three of those cards are doubled in a different suit (and therefore, whatever Y was told CAN NOT be conclusive).
3. Y: Now I know what this card is.: Player Y was told the rank and now knows the suit, so he knows the card.
4. Z: Now I also know what this card is.: This one is the kicker. If Player Y still does not know the card, then it is the 4 (since it shows up in multiple suits). Since Player Y DOES know the card, then the 4 is out. This leaves the A and the Q.
How to rule out the Ace or the Queen... I dunno. My 5 minutes is up, and it's time to read the other responses. Nice puzzle.

Edit: I was stuck on Hearts and forgot about the 5's. Damn!

Last edited: May 6, 2011