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 

Correct Address Identified !

How pointed towards the wrong one? 

Since Mr. House was able to run at one house after answers of passer-by, he must have got clear clues from that.

3. Is it square ?
  
First thing is sure that, the number must not be a non-square otherwise Mr.House wouldn't be sure as there are plenty of non-square numbers between 1 to 50. So it must be either 4,9,16,25,36,49. (1 is omitted for a reason)

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1. In which block is it?

Two possible answers here & 2 possible conclusions.

Block 1  :  4,9,16

Block 2  :  25,36,49

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2. Is the number even ?

Now had passer-by  answered Block 1 in 1st question & odd now then Mr. House would have come to know one exact number 9 (that's why 1 omitted here).

Or had he answered Block 2 in 1st question & even now then also Mr. House would have 1 number i.e. 36.

So in both cases, Mr. House would have got 1 fixed number with no point in asking extra question.


That means the passer-by must have told following answers & their possible conclusions are-

Block 1  :  Even  :  Square  :  4,16

Block 2  :  Odd   :  Square  :  25,49

Connect Dots with Straight Lines

Can you connect all nine dots with only four straight lines without losing contact with the paper while drawing? 

Read here how it can be done!

Source 
 

Connected Dots With Straight Lines

What was the challenge? 

This question often asked in personality development training courses. It needs some out of box thinking. In the question, no where it is mentioned that you line can't go beyond 3 dots. But our brain assumes that & try to find the solution according to that only!