# 28february

Guess the Day Problem Solution

In one small town, there is a liar who lies on six days of the week. But on the seventh day he always tells the truth. He made the following statements on three successive days:
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?

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