# Knights and knaves at a party

On an island, all inhabitants are either knights or knaves. Knights always tell the truth and knaves always lie. No tribe is gender related and some people marry into a different tribe.

When I visited this island, a tour guide who is a knight taught me an amazing question.

If you ask this question to a couple in the island, it is impossible to find out of which tribe he/she is or his/her partner is with a response from a single person. However, if you combine both answers from a couple, then you can figure out their tribes.

One day at a party, this amazing question was asked to 50 couples from the island. All husbands answered "Yes" and all wives answered "No". How many people lied?

What I think the answer is supposed to be:

1 person lied. Shout out to @Callidus for pointing out the somewhat obvious implication of my logic

My reasoning:

As I'll explain, there is no single question that can satisfy the requirements for the "amazing question". Hence OP lied in saying that such a question exists.

What I think the answer really is:

This question does not provide enough information for us to know how many people lied. As I show in my "alternative theory" section, there are compound questions that actually work. However, without knowing which question is used and the order in which the husband and wife are spoken to, we have no way to tell how many knights and knaves there are - you could formulate a question for which a husband saying "Yes" followed by the wife saying "No" means they are both knights while a wife saying "No" followed by the husband saying "Yes" means they are both knaves. So you could have 100 knaves, 100 knights, or any combination of knights and knaves.

There's a reason why the amazing question is not given to us:

No such real question could possibly exist unless it is asymmetrical!

Let's first tackle some properties of the question. We know that there are 4 possible pairings, KK, KN, NK, and NN (K=knight, N=knave). Based on the requirements of the question, we know that if you ask one person you get no conclusive information, but if you ask their spouse then you know both of their tribes. Let's focus on the husbands first:

Because we won't know his tribe based on his answer, any answer that he can give as a knight must be something he can give as a knave. Additionally, if his wife is of a particular tribe his response must change depending on his tribe. Because we won't know his wife's tribe based on his answer, his response must change depending on her tribe.

So if he's a knight, he'll say one thing (A) if she's a knight, and something else (B) if she's a knave. So if he's a knave, he'll say either A or B. Since he can't say A if she's a knight, he'll say B. Then, he'll say A if she's a knave.

H W | R
-------
K K | A
K N | B
N K | B
N N | A

The same logic holds for the wife:

H W | R
-------
K K | C
K N | D
N K | D
N N | C

Here's where we see the "amazing" part of the question:

H W | R R
---------
K K | A C
K N | B D
N K | B D
N N | A C

In order to satisfy the requirements that asking a single person gives you no information about their two tribes, we've forced the question to be in a form such that asking the spouse gives you no additional information. Asking once allows you to know whether or not the two are from the same tribe, but then you can gather no additional information by using the question.

In summary

This riddle can't be solved by attempting to derive the "amazing question", because it cannot exist. Since the question can't exist, there is no way to solve the puzzle in its current form - why should the husbands and wives have to give symmetrical answers? Perhaps in a KK couple, the husband will answer 'Yes' while the wife answers 'No'. Or perhaps that's what a NN couple would answer. So we could have either everyone telling the truth, or no-one.

An alternative theory

There are asymmetrical questions that make this work, such as having the answer change depending on whether or not you have talked to their spouse.

A possible truth-table for the question under these conditions:

1 2 | R R
---------
K K | Y Y
K N | N Y
N K | N N
N N | Y N

A question that fits this would be "If your spouse just answered my question did they say 'yes' otherwise if I asked you if you and your spouse are from the same tribe, would you say yes?"

Another possibility:

1 2 | R R
---------
K K | Y N
K N | N N
N K | N Y
N N | Y Y

A question that fits this would be "If your spouse just answered my question did they say 'no' otherwise if I asked you if you and your spouse are from the same tribe, would you say yes?"

Comparing the two possibilities:

This still doesn't help us solve the riddle. This allows the question to exist, but we have no way to determine how many people lied. If it is the second question and we always ask the husband first, then he will say "Yes" and his wife will say "No" if they are both knights (nobody lies). If we ask the wife first and then the husband, she will say "No" and he will say "Yes" if she is a knave and he is a knight (50 people lie). If we're not necessarily consistent in who we ask first, then we could have anywhere between 0 and 50 liars. It wouldn't be too hard to formulate another question so that the husband saying "Yes" and the wife saying "No" could either be two knaves or a knight and a knave, depending on who is asked first. Then we could have between 50 and 100 liars. In short, nobody or everyone could be liars depending on the order in which they are asked and on what the question is.

A note:

The reason I considered this an alternate theory instead of my main theory is because it feels like cheating - we're essentially asking two different questions. Also, if you object to the "if ... otherwise ..." construction of the questions, you can put it into a form that only uses 'and's and 'or's - "Is it true that 1. I have talked to your spouse and they said x, or 2. I have not talked to your spouse and if I asked you...". (might not be exactly the way to translate it, but I don't feel like going through it to make sure right now)

• Good summary. So how many people? :-) Feb 5, 2015 at 23:56

Assuming I have correctly digested Rob Watts' excellent answer, the number of people who lied is:

One.
There is no question that fits the description given. Therefore, a knight certainly would not have told you that one did, and you could not have asked it of 50 couples at a party. You made the whole thing up - you're the liar!

• Now that you've pointed it out, this seems totally obvious... +1 to you! Feb 6, 2015 at 16:58

50 people lied.

The reasoning:

In order to be able to tell nothing from one partner's response, but everything from both, not only must all combinations of answers be possible, but also a partner's answer must differ depending on the tribe of his/her partner. So the possible permutations should be of the form:

So,

Regardless of which response corresponds to Yes or No, we can tell that if each partner of a couple gives different responses to the question, they must be of different tribes. Given that for every couple, each partner gave a different response to the other, we can therefore see that all couples present must have been of a knight and a knave - therefore there must have been 50 knaves present, therefore 50 people lied.

Note that:

For the puzzle in its current form, it is not necessary to know what the amazing question was, or how the yes/no answers match up.

• Your table of responses doesn't quite work though (just like my attempt) e.g. a response of A tells you that person's spouse is a knave... Feb 5, 2015 at 18:33

50 people lied

And the question was:

Because

There are 4 types of pairs: KK, KN, NN and NK
Their corresponding answers to this question would be: YY, YN, NN, NY
Because the reply was YN (All husbands Yes and wives No), they are all KN (or NK) pairs and this 50 people were supposed to lie.

• But after you hear "No", you can be sure that the second person is a knight. Feb 5, 2015 at 6:39
• "If you ask this question to a couple in the island, it is impossible to find out which tribe he/she is or his/her partner is with a response from one person. However, if you combine all answers from a couple you can figure out their tribes." Otherwise answer can be 0 or 100 and the question could have been Has your partner answered me? or is your partner yet to answer? Feb 5, 2015 at 6:46
• With your other questions you will find out the tribe of the first person before asking the second. he/she is or his/her partner is, "or" is a really bad thing here. Feb 5, 2015 at 6:54
• @Xellos You are right Feb 5, 2015 at 7:00
• @MohitJain This "imaginary question" does not work. If you know the answer, then you can figure out what the first person is without asking the second. If you don't know the answer then it will not help you establish who is the knight and who is the knave.
– dmg
Feb 6, 2015 at 8:40

It's impossible to tell. Daniel's question "Are you from the same tribe as your spouse?" is one possibility for the amazing question (even though he got convinced otherwise in the comments). Two yeses mean both are knights; two nos mean both are knaves; a yes and a no means they are different tribes, and so the person who said yes is a knave. If this were the question then all husbands are knaves and all wives are knights, so the answer is 50.

But there are other, admittedly less natural, amazing questions. For example, we can ask "Do you have a husband of the same tribe or a wife of the opposite tribe?" This also works, but now if a husband says yes and his wife says no they are both knaves, so using this question the answer would be 100.

All the above assumes that each couple has one husband and one wife (as I imagine we are intended to, but OP doesn't actually say so). If same-sex marriages are possible then there could be any number of knights and knaves even using Daniel's question, since there would no longer need to be equal numbers of husbands and wives.

I could be off base here, but it seems to me the magic question could be "Are you and your partner of the same tribe?"

If you ask a knight married to a nave, the response is no. If you ask a nave married to a knight, the response is yes. If you ask a knight married to a knight, the response is yes. If you ask a nave married to a nave, the response is no. So both knights and naves could say yes, and both could say no.

But then you get the partner's response. Two naves each say no. Two knights each say yes. A night married to a nave says yes, while a nave married to a knight says no.

The question "are you and your partner from different tribes" also would work similarly.

If this is the question, 50 people lied.

• If you ask a knight married to a knave, the response is no. If you ask a knave married to a knave, the response is no. So if the answer is no, you can know that the partner of the answerer is a knave. Your magic question does not work. Apr 18, 2015 at 0:37
• Alas. I didn't think of that angle. Apr 18, 2015 at 0:46