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I have five close friends, Andrew, Benjamin, Cassie, Danielle and Eric. I know that one of them has a birthday today but I can't remember which.

I asked Andrew who had a birthday today, and he responded:

It's not my birthday today, but between me, Benjamin and Cassie exactly one of us is lying.

"Okay then." was my response.

I thought, "What exactly is that supposed to tell me?". I decided to go ahead and ask Benjamin whose birthday it was today. I wasn't too surprised when he replied:

It isn't my birthday today either. I am lying, or Cassie is lying or Danielle is lying, but only one of us is lying.

"But how am I supposed to know if you are telling the truth?" I asked. There was no response.

so I decided to ask Cassie instead, hoping for a more helpful response but expecting the worst. She told me this:

Out of all of us there are at least three liars

"Gee, thanks for that." I sarcastically responded. I was getting sick of this little game that my 'friends' were playing on me.

I had no choice, in order to find out whose birthday it was, I would have to get as much information as I could. This is what Danielle told me:

It is my birthday today!

At this point, I could not be sure, so I stuck with my plan and asked Eric what he thought. This is what he said:

What Cassie says is true.

I think that with that information, I should be able to figure out whose birthday it is, but I need you guys to help me out, so I can be sure. I am certain that if one of my friends lies, everything they say is false.

Whose birthday is it really?

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  • $\begingroup$ If someone is a liar and says something like "X and Y" or "X but Y", are we allowed (on the basis of "if one of my friends lies, everything they say is false") to infer that both X and Y are false, or only that "X and Y" is false? $\endgroup$ – Gareth McCaughan♦ Nov 29 '17 at 1:13
  • $\begingroup$ @GarethMcCaughan both must be false $\endgroup$ – micsthepick Nov 29 '17 at 1:22
  • $\begingroup$ Oh, I guess I should check: are we allowed to assume that it isn't the birthday of more than one of your friends? $\endgroup$ – Gareth McCaughan♦ Nov 29 '17 at 1:24
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    $\begingroup$ It bothers me a bit that Andrew has advance knowledge of your intention to go ask Benjamin and Cassie too; not to mention that he seems to know everybody's answers in advance. :-) $\endgroup$ – Bass Nov 29 '17 at 12:29
  • $\begingroup$ "It's not my birthday today, but between me, Benjamin and Cassie one of us is lying." Does this read as "at least one" or "exactly one"? I'd assume "exactly" as it seems to imply the sum (though with B specifying it in a more careful way we could easily see it the other way around), but currently it's not fully explicit. $\endgroup$ – Jakob Pamp Bengtsson Dec 4 '17 at 10:52
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As of 2017-12-05T03:30:04.412Z, there is one valid answer if only one friend has a birthday:

Danielle has the birthday.

This is because:

If Cassie was telling the truth, the three liars can't include her or Eric (because Eric says Cassie is telling the truth). This would cause the first two people to simultaneously have birthdays, which, in this half of my answer, is assumed to be impossible. Therefore, Cassie is lying. This means that there are only two liars. Since Eric always backs Cassie up, Eric is the other liar, and all others (including Danielle) tell the truth.

If multiple birthdays are allowed

Another scenario,

Andrew and Benjamin both have a birthday. (and possibly Cassie and Eric, we can't know)

Is also possible, and is internally consistent because:

A,B,D are lying. C and E are telling the truth because there are exactly 3 liars. D is lying, so it is not her birthday. Both pieces of Andrew's statement are wrong: is is Andrew's Birthday, and both him and Benjamin are lying. Both pieces of Benjamin's statement are wrong: it is his birthday, and both him and Danielle are lying.

All of this assumes that Benjamin only makes 2 claims.

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OP has clarified that "if one of my friends lies, everything they say is false" means that if someone lies and says "X but Y" then I can assume that both X and Y are false, and not merely that "X and Y" is false; and also that we aren't supposed to assume that only one friend can have a birthday today.

Then suppose

that Cassie is telling the truth. Then Eric is too, and there are at least 3 liars, so A,B,D are all liars. But one of the things A says is that one of A,B,C is lying (I take it A's statement isn't meant to imply that exactly one of the three is lying), which is in fact true in this scenario.

So that can't be the case, and therefore

Cassie is lying, and therefore so is Eric. This means there are at most 2 liars -- and we've found them. So A,B,D are all telling the truth. This means it's D's birthday and not A's or B's. How about the other statements? A says that one of A,B,C is lying: correct (C is). B says that exactly one of B,C,D is lying: correct again (once again, C is the liar).

So:

it's D's birthday; A,B,D tell the truth; C,E lie.

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  • $\begingroup$ sorry, it should say one of B, C or D. $\endgroup$ – micsthepick Nov 29 '17 at 1:31
  • $\begingroup$ Answer adjusted to match changed question. $\endgroup$ – Gareth McCaughan♦ Nov 29 '17 at 1:37
  • $\begingroup$ solid logic, but note that the solution is not dependant on there being one person with a birthday. $\endgroup$ – micsthepick Nov 29 '17 at 1:47
  • $\begingroup$ OK, answer adjusted so it doesn't depend on that assumption. $\endgroup$ – Gareth McCaughan♦ Nov 29 '17 at 2:31
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I think I found an easy solution.

If E lies, C also lies, which means there are less than 3 liars which means A, B and D are truth tellers so it is D's birthday.
If E tell the truth, so does C which means that A, B and D are liars which means both A and B have birthday today, which is impossible.
So it is D's birthday today. (He told you so to your face! Why couldn't you trust your friend! :P)

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  • $\begingroup$ if E and C lie, couldn't there still be another liar? you need to explain why in order for this answer to work. $\endgroup$ – micsthepick Dec 6 '17 at 2:37
  • $\begingroup$ @micsthepick If C lie, 3 or more liars is a lie, so 2 or less is the truth. We already have 2 liars so that's it. $\endgroup$ – stack reader Dec 6 '17 at 3:08
  • $\begingroup$ 3 or more liars would not still be a lie if there were more liars. You need to show why that can’t be still. $\endgroup$ – micsthepick Dec 6 '17 at 3:12

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