# How many of them are lying?

Here is a puzzle I am struggling to solve:

Seven people are in an argument, but potentially some or all of them are liars. They give the following statements:

Bob: "No one lies."

Jennifer: "No one tells the truth."

Conrad: "Jennifer is not a liar."

Tom: "Conrad and Sherry always lie at the same time."

Sherry: "Danny never lies."

Danny: "Sherry is a liar."

How many of them are lying?

In everyday life I am usually quite good at this, however here I am buffed on where to start from... I think that once I have a starting point, everything will flow smoothly afterwards.

• Are we assuming that there's no possibility that some of them are simply mistaken and not lying? – Ian MacDonald Sep 6 '16 at 11:48
• Well the question is how many of them are lying, so I guess we cannot assume that anyone is not lying. In regards to the mistaken part, I guess we assume their are either lying or not lying, but nothing in the middle (i.e. mistaken). – Newskooler Sep 6 '16 at 11:52
• As you will see from the answers here, we cannot tell how many are lying because the truth values of the statements made can't all be determined (nor e.g. are the undetermined ones such that we can tell exactly one of P,Q is true even though we don't know which). I think this is a poorly-constructed puzzle. – Gareth McCaughan Sep 6 '16 at 11:56
• is a person a liar if he/she lies only sometimes? – Marius Sep 6 '16 at 12:00
• I read through the answers and I also read the Liar's Paradox, as this seems to be going on here. As I understand, the Liar's Paradox is actually inconclusive by definition (it's like watching yourself in the mirror, with mirrors behind you as well). So, as I have no experience in such puzzles, to me it seems that there is no logical ground to stand on and hence I too agree it may possibly be badly constructed. Or there is a logical ground? Even from the start, Bob, Jennifer and Conrad are falling in the Liar's Paradox... – Newskooler Sep 6 '16 at 12:02

Bob: "No one lies."

It can be true if and only if (iff) all other statements are true.

Jennifer: "No one tells the truth."

It can be true iff all other statements are lies.
Thus, Bob is definitely lying.
Also, she must not be speaking truth; because by universal quantification, 'Nobody speaks truth' means she is also not speaking the truth.

Conrad: "Jennifer is not a liar."

As shown above, Jennifer is indeed a liar. Thus this is not true and Conrad is also a liar.

Tom: "Conrad and Sherry always lie at the same time."

If this is true, Sherry is also a liar from the previous one. If this is false, Sherry is not a liar. We'll deduce it from further statements.

Here comes the little loop...
Sherry: "Danny never lies." (let's call it statement S)
Danny: "Sherry is a liar." (let's call it statement D)

S can be true iff Danny always speaks truth (assuming that Danny says something and not keeps silent).
If D is a true statement, then it would be ok (Sherry is lying about Danny never lying). But if it is a lie then it becomes unclear, whether Sherry is a liar and/or Danny is a liar.

Then comes the last but not the least :

Again, we cannot quantify sometimes. If this is true, then Danny is also a liar(the one who doesn't always lie)! But, he is not lying in this case.
And as it goes, in statement D, Danny speaks truth. Which must make Sherry a liar.
Also, this means Tom is not lying.

$4$ people are lying in the given statements.

Lying:

Not Lying:

Danny(in the given statement), Adam, Tom

• How do you know that Adam and Tom aren't lying? – Sconibulus Sep 6 '16 at 12:47
• Just because Sherry and Conrad are both lying does not mean Tom is telling the truth, they might not always lie at the same time. – Jonathan Allan Sep 6 '16 at 12:51
• @JonathanAllan I understand that, but this is what he have on Tom. The words, 'sometimes', 'always' cannot be used interchangeably. I am assuming the statements given contain all the information to check the always... – ABcDexter Sep 6 '16 at 12:55
• @ABcDexter I think you can be sure Adam isn't lying, since his statement is the exact opposite of Sherry's and she is definitely a liar. Tom is still a wildcard, though. – trentcl Sep 6 '16 at 18:01
• @ABcDexter Good point! At first, I considered Sherry's statement, when false, to mean "Danny sometimes lies", but as you point it out could also mean he always lies; that fulfills the critera. But should we consider "sometimes" as a subset of "always"? If not, Adam's statement isn't necessarily true. I'd also say we can't deduce if Tom's statement is true or not. We don't know about "always", only "in this instance". I'd chalk it down to Tom and Adam being unknown from what we have, Adam depending on sometimes/always keywords. I agree with the "at least 4" answer. – Jakob Pamp Bengtsson Sep 8 '16 at 10:39

If Bob's statement, "No one lies", were true then it would make Jennifer's statement, "No one tells the truth", a lie which it cannot be if no one lies. So Bob is a liar.

If Jennifer's statement, "No one tells the truth", were true then it would make her own statement a lie. So Jennifer is a liar.

Danny and Sherry refer to each other, but if Sherry's statement "Danny never lies" is true then Danny is telling the truth when he says "Sherry is a liar", so Sherry is lying and Danny is telling the truth.

Tom's statement "Conrad and Sherry always lie at the same time" is true in the current state, but we cannot tell if it is always the case.

Adam's statement "Danny sometimes lies" could, in isolation, also be true or false since Danny is telling the truth at the moment. However, we know Sherry is lying so Danny does lie sometimes and Adam must therefore be telling the truth.

So Liars are Bob, Jennifer, Conrad, and Sherry;
Danny and Adam are telling the truth;
Tom could either lying or telling the truth.

Leading to $2$ possible states with either $4$ or $5$ liars:

B, J, C, T, S, D, A
-------------------
L, L, L, L, L, T, T
L, L, L, T, L, T, T
If one simply changed Tom's statement to "if one of Conrad and Sherry are lying then both are", "Either Conrad and Sherry are both lying or both are telling the truth", or some other equivalent statement then the puzzle would have a single solution (the bottom one, with $4$ liars).

• The assertion that leads to your Option 1 is unsound: "No one tells the truth" and "No one lies" can both be false if Jennifer and Bob are liars but someone else is a truthteller; there is no paradox. – trentcl Sep 6 '16 at 17:41
• @trentcl How very true! I shall remove it. – Jonathan Allan Sep 7 '16 at 5:44

Are all liars, their statements are mutually incompatible.

Sherry and Danny

Have mutually incompatible statements, Sherry's cannot be true so Danny's must be.

Tom

Can't be determined. In this case, both Bob and Sherry are liars, but we don't know the larger trend.

Also can't be determined, it's a sometimes clause, where we have one data point.

All in all

There are at least 4 lies, and not more than 6.

Jennifer is clearly a liar. If not, then it means nobody tells the truth including her, so, boom, brain explodes. Paradox.
This means that Bob is also a liar (because we established Jenny is a liar).
This leads to Conrad being a liar, because he takes Jennifer's side. Is there something going on between the 2 of them?

Now it gets tricky.

Let's take Sherry and Danny. If Sherry is right, this makes Danny a liar which makes Sherry wrong. Boom, brain explosion again.
This means that danny lies but only sometimes, but in this case he tells the truth.
These also fit with Tom's statement. So Tom might be telling the truth.

Conclusion:

Bob: Liar
Jennifer: Liar
Tom: Inconclusive. Could be both. This statement matches this time, but don't trust Tom just yet.
Sherry: Liar
Danny: Normal person. Sometimes lies sometimes he tells the truth.