Exactly one of Sam or Tom has family visiting. They make these statements:

Sam: "Tom has family visiting, but I don't."

Tom: "Sam is lying, or I am lying, or possibly we're both lying."

Which of the two men has family visiting?

Attribution: this puzzle is from The Mensa Puzzle Calendar, July 17, 2018. I solved it as intended but the solution doesn't sit right with me. I want to see what other solutions can be found.

  • $\begingroup$ Yay! A user other than me finally has a big pie on their profile picture, from what I've seen :D $\endgroup$ – Mr Pie Sep 8 '18 at 13:22
  • $\begingroup$ Max, how do these solutions look in terms of adding clarity? Or were you thinking of a different way to show the answer altogether? $\endgroup$ – El-Guest Sep 8 '18 at 13:30
  • $\begingroup$ @El-Guest the calendar gives a very thorough explanation of its answer (which you found). But I suspect that there are other answers. $\endgroup$ – Max Sep 8 '18 at 13:35
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    $\begingroup$ Without more meta-information (such as: "one of Sam or Tom always tells the truth"), this isn't solvable or even a puzzle. Imagine that Sam only ever says this one sentence, and ditto for Tom. Either of them might have family visiting to listen to their nonsense for a while. $\endgroup$ – Eric Tressler Sep 8 '18 at 16:43
  • $\begingroup$ Yeah, that was my thinking too. I was very disappointed that the calendar's explanation of it's answer assumed such meta-information. $\endgroup$ – Max Sep 9 '18 at 6:27

If Tom is lying, that makes his statement true (a contradiction), so Tom must be telling the truth. This means Sam must be lying. Since exactly one of the men has family over, it must be Sam since otherwise Sam would be telling the truth.

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    $\begingroup$ El-Guest answered before you, but he completely rewrote it after your answer, so you get my upvote. $\endgroup$ – Max Sep 8 '18 at 13:34

Update because I’m a dummy and it’s too early for my thinking brain:

I think that it is

Sam who has family over.


Tom’s statement is true (he’s a truth teller) if either he is lying or if Sam is lying. Tom’s statement is false (he’s a liar) if both he and Sam tell the truth. The second scenario leads to a contradiction, because he needs to tell the truth in order to be a liar. Tom’s statement must therefore be true; and so Sam must be lying. It is therefore Sam who has family over.

  • $\begingroup$ damn it was a bit too slow :D $\endgroup$ – Kevin L Sep 8 '18 at 13:27

This is not the Mensa-approved answer, but

The question never specifies that Sam and Tom give logically consistent information such as only telling complete truths or lies. Both of them could be spouting gibberish that has nothing to do with which one has family visiting. So the answer is that it is impossible to know which has family visiting.

  • $\begingroup$ Right? I mean, yes, I understand the "logic" of the approved answer, but imagine if Tom had just said "Sam is lying". Apparently adding "or I am lying, or both" makes Mensa members trust him ... like "I am the Pope"... hmm really ... "I am the Pope, or I am lying, or both" ... oh well, in that case you must be the Pope $\endgroup$ – deep thought Oct 11 '18 at 4:11

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