Let's assume all three things he says is a lie and see if that's consistent:
- Day1: If this is a lie, then he tells the truth either on Monday or Tuesday.
- Day2: If this is a lie, then it must be Mon, Tue, Wed, or Fri.
- Day3: If this is a lie, then he must tell on the truth on Wed or Fri.
Clearly this is not consistent, because Day1 and Day3 cannot both be lies (likewise they cannot both be the truth since he only tells the truth one day of the week). So now, just assume one is the truth, let's assume Day1, he's telling the truth and Day3 he's lying.
- Day1: If this is the truth, then we know for sure he lies on Mon and Tue.
- Day3: If this is a lie, then he tells the truth either on Wed or Fri
- If we assume Day1 is the truth and Day3 a lie, then Day1 is either Wed or Fri meaning that Day2 is either Thu or Sat. This is inconsistent with Day2 being a lie.
This means that Day1 must be a lie and Day3 must be the truth. If Day1 is a lie, then he tells the truth on Mon or Tues which makes Day3 either Mon or Tues, which makes Day2 either Sun or Mon. Day2 cannot be Sun, but it could be Mon (since he definitely lies on Day2). This means:
- Day1: Sun (a lie).
- Day2: Mon (a lie).
- Day3: Tues (the truth).
So he tells the truth on Tuesday.