Truth Of a Strange Liar

What was his story? 

To find the truth we need to logical deduction here.

Now if statement on Day 1 is untrue then Richard must be telling the truth on Monday or Tuesday. 

And if Day 3 statement is untrue then he must be telling the truth on Wednesday or Friday. 

But he speaks true only on 1 day. So both statements of Day 1 & Day 3 can't be true at the same time. If so, then Richard speaks true on 2 days either Monday/Tuesday or Wednesday/Friday. This means that one of statements from Day 1 must be true & other must be untrue. That also makes the statement on Day 2 untrue always.

Case 1 : Day 3 statement is untrue.

In this case, Richard must be telling truth on either Wednesday or Friday. The statement on Day 1 would be true according to above logical deduction. Hence Day 2 must be either Thursday or Saturday. In both cases, statement on Day 2 would be true.

Case 2 : Day 1 statement is untrue.

If the statement made on Day 1 is untrue then Richard tells truth on Monday or Tuesday. Other statement on Day 3 must be true means Day 3 must be either Monday or Tuesday. If so, then Day 2 must be either Sunday or Monday. In case of Sunday, Day 2 statement would be true & in case of Monday Day 2 statement would be untrue. Hence Day 2 must be Monday & Day 3 must be Tuesday.

So Richard tells truth on Tuesday.

Locker Room & Strange Principal

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:

There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has asked the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?

What principal was trying to teach? 

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