Day 1: 'I lie on Monday and Tuesday.'

Day 2: 'Today, it's Thursday, Saturday, or Sunday.'

Day 3: 'I lie on Wednesday and Friday.'

What day does the guy tell the truth?

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Solution:

.. He speaks the truth on Tuesdays.

The assertions of Day 1 and Day 3 can't both be true.

Else there would be two truth days: Day 1 and Day 3.

So he cannot lie on all four days mentioned: M T W and F.

The truth day is M T W or F.

The assertions of Day 1 and Day 3 also can't both be false.

Else the truth day would be M or T and also would be W or F, making two truth days.

The truth day is Day 1 or Day 3.

The assertion of Day 2 can't be true.

Else there would be two truth days: Day 2 and Day 1 or Day 3.

Negating Day 2's statement, we see that

Day 2 is M, T, W or F.

Day 2 can't be T or W.

Else Day 1 would be M or T, on which days he claims to lie. A paradox.

Day 2 can't be F.

Else Day 1 would be Th on which he claims to lie M T, and Day 3 would be Sat on which he claims to lie W F.

This makes two truth days: Day 1 or Day 3 [Thu or Sat] and M, T, W or F.

Day 2 is M.

The truth day is Sun or T.

The truth day can't be Sun.

Else, on Day 3 he lies about lying on both W and F, creating two truth days: Sun and W or F.

The truth day is T.

Check:

Day 1 [Sun] I lie on M T. [a lie - he speaks truth on T] - OK

Day 2 [Mon] It's Th, Sat or Sun. [a lie - it's M] - OK