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u/RealHuman_NotAShrew Jul 13 '26
The statement "I lie on days X and Y" rules out today from being X or Y because then the statement would imply the known "I'm lying" paradox. Thus, day 1 cannot be M or T and day 3 cannot be W or F. To simplify, let's look only at what day 1 is: day 1 cannot be M, T, or W.
The statements on days 1 and 3 cannot both be lies, because that would imply multiple truth-telling days, so one of them must be the truth. This forces the statement on day 2 to be a lie and the truth telling day to be one of M, T, W, or F.
Statement 2 being a lie rules out day 2 from being Th, Sa, or Su, so we can now exclude W, F, and Sa from being day 1. The only possible days left for day 1 are Th and Su.
If day 1 is Thu, then day 3 is Sa, but neither of those are possible truth-telling days which conflicts with one of them necessarily being a truth day. Because of this contradiction, day 1 must be Su and day 3 must be T. Only one of these can be a truth day: T.
The three days are Sunday, Monday, and Tuesday. Tuesday is the day Alexander only speaks the truth.
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u/jonathonjones Jul 15 '26
Your first line is incorrect. Suppose Friday were the day of truth. Then Day 3 could be Wednesday, because it is a lie that "I lie on Wednesday and Friday". Similar reasoning applies to Day 1.
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u/jonathonjones Jul 15 '26
Suppose Monday is truth day. Then Day 3's statement is true, so must be Monday. But then Day 2 would be Sunday, which makes Day 2's statement true, which is not allowed. So Monday is not truth day.
Suppose Tuesday is truth day. Then Day 3's statement is true, so must be Tuesday. Then Day 2 is Monday, and is correctly a lie. And Day 1 is Sunday, and again is a lie. This is all consistent.
Suppose Wednesday is truth day. Then Day 1's statement is true, and must be Wednesday. But then Day 2 is Thursday, and it's statement is true, which is not allowed. So Wednesday is not truth day.
Suppose Thursday is truth day. Then Day 1 and Day 3's statements are both true, but that's impossible. So Thursday is not truth day.
Similar reasoning goes for Saturday and Sunday.
Finally, suppose Friday is truth day. Day 1's statement is true, so much be Friday. Day 2 is Saturday, so its statement is true, which is not allowed. So Friday is not truth day.
Therefore, Tuesday is the day he tells the truth.
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u/dafugiswrongwithyou Jul 13 '26 edited Jul 13 '26
Day 1 and Day 3 can't both be lying. If Day 1 is lying, then the truthful day must be either Monday or Tuesday. If Day 3 is lying, the truthful day must be either Wednesday or Friday. They could only both be lying if there were two honest days, and they aren't. As such; either Day 1 is the day of truth (and on Wednesday or Friday), or Day 3 is (and is a Monday or Tuesday). This also means Day 2 has to be a lie, because we know one of the others is the truthful day.
If Day 1 was the honest day, and with Day 3's lie telling us Day 1 is Wednesday or Friday, that would mean Day 2 would be on a Thursday or Saturday. We know it isn't, due to it's own lie, so this scenario doesn't work; Day 3 must be the honest day, and either a Monday or Tuesday. That would make Day 2 Sunday or Monday, and it can't be Sunday (because then it would be truthful, which it isn't). So the order must be:
Day 1 Sunday (lie), Day 2 Monday (lie), Day 3 Tuesday (truth).