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.
Answer
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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