A Strange Liar

Richard is a strange liar. He 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."

On which day does Richard tell the truth? 

  Am I a liar?

Find the truth here! 

Source 

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? 

Source 

The Lesson Taught By Strange Principal

Where it did begin? 

While finding the solution we need to keep basic fact from the problem in mind. Since lockers were closed initially, the lockers which are 'accessed' for odd number of times only are going to open. Rest of all would be closed.

Now task is to find how many such lockers are there which were 'accessed' for odd number of times.

Let's take any number say 24 for example, which is not perfect square & find out how many factors it has.

24 = 1 x 24
24 = 2 x 12
24 = 3 x 8
24 = 4 x 6

So factors are 1,2,3,4,6,8,12,24 i.e. 8 numbers as factors which is even number. Every factor is paired with other 'unique' number! So this pairing always makes number of factors 'even'. In the problem, this lock no.24 will be 'accessed' by 1st, 2nd, 3rd..................24th student. That means 'accessed' even number of time & hence would remain closed.

Now let's take a look at lock no. 16 in which 16 is perfect square. Finding it's factors,

16 = 1 x 16
16 = 2 x 8
16 = 4 x 4

we get 1,2,4,8,16 i.e. 5 numbers as factors which is odd. The reason behind is here 4 appears twice (with itself) while rest of others are paired with other 'unique' number. Hence, number of factors of a perfect square are always odd. Now here lock 16 would be accessed by 1st, 2nd, 4th, 8th, 16th i.e. 5 times. Hence it will be open.

Like this way, every lock with number which is perfect square would be 'accessed' for odd number of times & hence would remain open! e.g. 1,4,9,16,25,49 & so on.

Now 961 (31^2) is the maximum perfect square that can appear within 1000 (32^2) as 1024 goes beyond.

Hence there would be 31 locks open while rest of all closed!

Lesson Of The Day

So what lesson taught by strange principal? The number which is perfect square has odd number of divisors.

 

Unlock The Distance

Distances from you to certain cities are written below.

BERLIN = 200 miles
PARIS = 300 miles
ROME = 400 miles
AMSTERDAM = 300 miles
CARDIFF = ?? miles

How far should it be to Cardiff ?

 How far? Find Here! 

Source 

The Distance Unlocked

What was the question?

Just count Vowels V & Consonants C in any 2 spelling to get how much they value.


From BERLIN,

2V + 4C = 200

V + 2C = 100             ........(1)

From ROME,

2V + 2C = 400

V + C = 200               .......(2)

Solving (1) & (2), we get,

V = 300 & C = -100

For CARDIFF, we have,


2V + 6C = 100. 

So CARDIFF = 100 miles

Wrong Address By Liar

Mr. House would like to visit his old friend Mr. Street, who is living in the main street of a small village. The main street has 50 houses divided into two blocks and numbered from 1 to 20 and 21 to 50. Since Mr. House has forgotten the number, he asks it from a passer-by, who replies "Just try to guess it." Mr. House likes playing games and asks three questions:

1. In which block is it?

2. Is the number even?

3. Is it a square?

After Mr. House has received the answers, he says: "I'm still doubting, but if you'll tell me whether the digit 4 is in the number, I will know the answer!". Then Mr. House runs to the building in which he thinks his friend is living. He rings, a man opens the door and it turns out that he's wrong. The man starts laughing and tells Mr. House: "Your advisor is the biggest liar of the whole village. He never speaks the truth!". Mr. House thinks for a moment and says "Thanks, now I know the real address of Mr. Street".

 
What is the address of Mr. Street?

This is how Mr.House found correct address! 

Source