Logic Question

Factual Questions
1

ok here we go~
I have a set of cards and 2 people. I will give each person one card and they will be in consecutive order. For example If person A was to get 3 person B will either have 2 or 4. This is the same with all numbers.

I then ask Person A if he knows what card person B has he says “No”. I then ask Person B if he knows what card A has he says “NO” I ask A again and he says “NO” then when B is asked again he replys “yes”

How is this possible?
I would really appreciate an answer to this
THank you.

2

Let me assume that the sequence of cards is A,2,3,…,10,J,Q,K. Also, that suits don’t matter, and that A,K are not “consecutive”, and that A and B know what the other’s statements are.

Suppose A has an A or K. He would then know what card B has (either 2 or Q). But A says “no” so we (and B) can conclude that A does not have A or K. Knowing this, when B says “no”, he must not have a 2,Q,A, or K (else he would know what A has). Knowing this, when A says “no” a second time, we conclude that he doesn’t have a 3 or J either. Now, when B says “yes” it must be because he holds either a 4 or 10. The 4 is possible, because we’ve eliminated the possibility of A having a 3, then B knows A holds a 5. Similarly for the 10, (A has 9).

I think this fits your scenario.

3

the problem with this would be that if he person had an A the other person would not be able to have a 5 because they will not be consecutive.

4

bbalkingnyc:

I think you misunderstood the explanation. In the specific example you gave either person B has a 4 and person A has a 5 or person B has a 10 and person A has a 9. The only assumption made that is not given is that the two people do know what number they themselves have.

5

Person B is lying or,
Person A revealed his card to person B or,
Person B saw person A’s card over his shoulder.

Seriously, I think you missed out some vital conditions in your problem:

Each person has no way of seeing the other’s card
The cards are a standard set of playing cards with Joker’s removed.
Whether or not you count through J, Q, K, A, 2 as a consecutive run.

etc. etc.
I’ve probably missed a few. But personally, I still think Person B is a lying rat. :slight_smile:

6

First round:

If A has A, he knows that B has 2.
If A has K, he knows that B has Q.
If A has any other card, he doesn’t know what B has.
A answers “no”, and therefore has some card from the set {2,3,4,5,6,7,8,9,T,J,Q}.

Second round:

B cannot have 2 or Q because A answered “no”.
If B has A, he knows that A has 2.
If B has K, he knows that A has Q.
B answers “no”, and therefore has some card from the set {3,4,5,6,7,8,9,T,J}.

Third round:

A cannot have 3 or J because B answered “no”.
If A has 4, he knows that B has 5.
If A has J, he knows that B has T.
A answers “no”, and therefore has some card from the set {4,5,6,7,8,9}.

Fourth round:

B cannot have 4 or 9 because A answered “no”.
If B has 5, he knows that A has 6.
If B has 8, he knows that A has 7.
That leaves A having some card from the set {5,8}.
If B has 6, he knows that A has 5.
If B has 7, he knows that A has 8.

B now knows what card A has and answers “yes”.

7

:rolleyes:

That’s what happens when typing rhythm overtakes pattern recognition…

First round:

If A has A, he knows that B has 2.
If A has K, he knows that B has Q.
If A has any other card, he doesn’t know what B has.
A answers “no”, and therefore has some card from the set {2,3,4,5,6,7,8,9,T,J,Q}.

Second round:

B cannot have any card from the set {2,Q} because A answered “no”.
If B has A, he knows that A has 2.
If B has K, he knows that A has Q.
B answers “no”, and therefore has some card from the set {3,4,5,6,7,8,9,T,J}.

Third round:

A cannot have any card from the set {A,2,Q,K} because B answered “no”.
If A has 3, he knows that B has 4.
If A has J, he knows that B has T.
A answers “no”, and therefore has some card from the set {4,5,6,7,8,9,T}.

Fourth round:

B cannot have any card from the set {A,2,3,4,T,J,Q,K} because A answered “no”.
If B has 4, he knows that A has 5 (else A would have said “yes” for 3)
If B has T, he knows that A has 9 (else A would have said “yes” for J)
If B has 5, he knows that A has 6.
If B has 9, he knows that A has 8.

Otherwise, A has 7.

8

This first assertion in the second round is false. The only thing A’s answer of “no” tells B is that A does not have A or K. If A had 3, he would have to say “no”, but B could still have a 2.

It seems obvious to me that you’re on the right track, but you’ve made an error. I’m still thinking through it myself…

9

Okay, here’s what I think, but I have to assume a few things that weren’t made clear in the original problem.

First, I am assuming the same as everybody else – standard card deck with a low to high range, no wraparound (I’ll assume Ace is the low card, like the other examples here).

Second, I am assuming that the persons A and B are telling the truth.

Third, I am assuming that the question “How is this possible?” is looking for a single case in which this scenario could be true (i.e. NOT necessarily true with every pair of cards).

Based on this, here is one situation where I would say this is possible. Person A has a 4, person B has a 3.

Arguments:

  1. Person A knows that B has a 3 or a 5, but doesn’t know which.
  2. Person B knows that A has a 2 or a 4, but at the start, doesn’t know which.
  3. Person A has to answer “no” both times, because there are really no true clues for him from B’s single “no” response.
  4. A’s first “no” response doesn’t tell anybody much of anything, since B already knows that A doesn’t have the Ace (and B already expected a “no” answer here).
  5. B’s first “no” response also doesn’t tell A anything, but it will tell B something, as soon as A responds again. And here’s why:

After B responds ‘no’ (and remember he knows A has 2 or 4), here is B’s mental thought process:
“Case I: A has a 2, then my ‘no’ response would have told my worthy opponent that I did not have the Ace. So now A knows I have the 3, and will answer YES next.
** Case II:** A has a 4, so my ‘no’ response won’t tell him a thing, and he will have to answer NO next.”

  1. Since A’s next answer was “no”, B knows that it’s Case II. And thus he now knows A has the 4.
10

Two more key assumptions which have only been alluded to (Monstre mentioned “mental thought process” and “worthy opponent”) is that the players are ‘reasonably intelligent’ and rational – in other words they have the ability to reason as well as we can assume, and that they are interested in the outcome (i.e. they want to try to find the answer, even if it means their opponent could win).

Obviously, someone slow-witted or unconcerned could just sit there saying, “I don’t know” even if they are holding a 1 (or an A).

Those with some ability can come up with an eventual answer as already described, as long as all card numbers are different and bounded on at least one side (like whole numbers, or playing cards as already described).

If anyone’s still having trouble getting it, there’s a good explanation (with diagrams) from the Car Talk site (this was the weekly puzzler from a few months back).

Here it is : Car Talk Magic Hat Answer

There’s a bunch of similar kinds of puzzles (those which require “reasonable intelligence” of the participants) involving men with and without hats somewhere in one of Martin Gardner’s books.