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

------------------------------------------------------------------------------------------

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!

The Alphamatic Problem

In the addition below, all digits have been replaced by letters. Equal letters represent equal digits and different letters represent different digits.

   ABCABA 

+

   BBDCAA 

+

   ABEABB 

+ 

   ABDBAA

-------------------

 AAFGBDH

What does the complete addition look like in digits? 

Note : Alphamatic in the title is word derived from Alphabets & Mathematics. In such problems numbers are replaced by alphabets. The challenge is to find the number for each alphabet satisfying given mathematics equation.

Step by step answer is here! 

Source 

Answer Of Alphamatic Problem

Here is question!

 First of all let's write down the equation once again.


   ABCABA 

+

   BBDCAA 

+

   ABEABB 

+ 

   ABDBAA

-------------------

 AAFGBDH

We will refer to places in number from left as a first, second, third...sixth instead of tenth, hundredth, thousandth etc. 

First we need to find if the 5 digits of first number itself i.e. ABCAB are carries forwarded from previous place.  

From the addition of variable from first place, we get,

3A + B  = 10A + A ........(1)

B  =  8A              ........(2)

Only numbers satisfying above are A = 1 & B = 8 , but at previous place we have addition of 4 B's. If B = 8, then addition at second place would be 32 with F = 2 & carry 3 which is not equal to A = 1. So A can't be a carry. So we need to modify (2) above as

B + x = 8A .........(2)....Where 'x' is carry forwarded from second place.

If B = 1 or 2 then x = 0 as  at second place we would have F = 4 or 8. In that case, A would be fractional. Some other possible combinations for B, A & x are,

B = 9, x = 3,   8A = 12,

B = 8, x = 3,   8A = 10,

B = 7, x = 2,   8A =  9,

B = 6, x = 2,   8A = 8,

This is the only combination that can make A a whole number. So A = 1, B = 6.

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From sixth place, we have,

H = 3A + B = 9

---------------------------------------------------------------------------------------------

From fifth place, we have,

D = 2A + 2B  = 14

But it has to be single digit i.e. D = 4 with 1 carry forwarded to next.

---------------------------------------------------------------------------------------------

From fourth place,

B = 2A + B + C + 1  .....1 is carry from last place.

6 = 9 + C

Now C can't be negative hence C + 9 has to be 2 digit number with 6 at last digit.Since addition of 2 single digit numbers never exceeds 18, C + 9 has to be 16.

16 = 9 + C 

gives, C = 7 & carry 1 forwarded to third place.

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Nightclub Tragedy

An American nightclub called 'The Coconut Grove' had a terrible fire in which over 400 people died. A simple design flaw in the building led to the death toll being so high. Subsequently, regulations were changed to ensure that all public buildings throughout the country eliminated this one detail which proved so deadly.

What was it?


Get here to know what was that! 

Source 

Change In Regulations After Tragedy

What was the tragedy? 

This is based on a true story:

The doors at the Coconut Grove opened inward. In the mad dash to escape the fire, people were crushed against the doors and couldn't pull them open. After the Coconut Grove disaster in 1942, all public buildings had to have doors which opened outward.

Loss Due To Fake Currency Note?

A lady buys goods worth Rs. 200 from a shop. (shopkeeper is selling the goods with zero profit). The lady gives him Rs. 1000 note. The shopkeeper gets the change from the next shop and keeps Rs. 200 for himself and returns Rs. 800 to the lady. Later the shopkeeper of the next shop comes with the Rs. 1000 note saying "duplicate" and takes his money back.
 
How much LOSS did the shopkeeper face?

Find that LOSS here!

Source 

'Cashless' Transaction

What was the tricky? 

The shopkeeper's getting change for Rs.1000 from the next shop is just to confuse you.
He brought Rs.1000 from that shop & gave back Rs.1000. There is no loss or profit from that end. Now he gave genuine Rs.800 & product of Rs.200 to the lady. In return, he got Rs.0 as the Rs.1000 note offered by the lady was fake one. So total loss the shopkeeper faced is of Rs.1000. 

 

Same Issue Different Approaches

Two newly launched firms started manufacturing soaps in their production unit. After few day, both started facing the same issue. Few soap wrappers were remaining empty without soaps within those. Both manufacturers asked their employees to find solution on this. Employees of one firm did lot of research & developed a machine to detect the empty wrappers. For that they invested lot of time & money. While employees of competitor were smart & just brought one thing from the market & solved the problem.
What was that thing? 

Know here what did they bring? 

The Most Efficient Approach Matters

What was the issue? 

The employees of other firm were smart. They just brought table fan from the market. Now they put it in front of chain where soaps with wrappers are placed & moved for the packaging at last stage of production. Air flowing from fan started flowing away empty wrappers off the chain. In this way, they invented smart way to detect the empty wrappers. A simple solution that saved a lot of money & time! 

"Make The Line Shorter!"

Once a teacher asked student the question that Akbar once has asked to Birbal. Teacher drawn a line on a paper with pencil & posted paper on a board. He asked students to make the line shorter without erasing by eraser or extending it with pencil.

One of the student who knew that story of Akbar - Birbal came & just draws another bigger line ahead of previous line. Now the other line looked shorter.

   
Now teacher decides to trick the students. 'Now without touching any line it make left line longer & right line shorter', he asks further.

The student was smart & just rotated the paper upside down. That way, now left line looked longer & right one shorter!

Complex Time, Speed & Distance Maths

There is a circular race-track of diameter 1 km. Two cars A and B are standing on the track diametrically opposite to each other. They are both facing in the clockwise direction. At t=0, both cars start moving at a constant acceleration of 0.1 m/s/s (initial velocity zero). Since both of them are moving at same speed and acceleration and clockwise direction, they will always remain diametrically opposite to each other throughout their motion.

At the center of the race-track there is a bug. At t=0, the bug starts to fly towards car A. When it reaches car A, it turn around and starts moving towards car B. When it reaches B, it again turns back and starts moving towards car A. It keeps repeating the entire cycle. The speed of the bug is 1 m/s throughout.

After 1 hour, all 3 bodies stop moving. What is the total distance traveled by the bug?

Find out complex calculations here!

Simple Solution of Complex Problem

Here is that complex looking problem! 

Everything built, written or designed in the given problem is to distract you from basic physics formula.

Speed = Distance / Time

Hence,

Distance = Speed x Time 

All the details given except speed of bug & time for which it traveled are there to confuse you. Speed of bug is 1 m/s & it traveled for 1 hour = 3600 seconds.

Distance = 1 x 3600 = 3600 m

So the total distance traveled by the bug is 3600 m.

 

What was the neighbor's solution?

An old farmer died and left 17 cows to his three sons. In his will, the farmer stated that his oldest son should get 1/2, his middle son should get 1/3, and his youngest son should get 1/9 of all the cows. The sons, who did not want to end up with half cows, sat for days trying to figure out how many cows each of them should get.

One day, their neighbor came by to see how they were doing after their father's death. The three sons told him their problem. After thinking for a while, the neighbor said: "I'll be right back!" He went away, and when he came back, the three sons could divide the cows according to their father's will, and in such a way, that each of them got a whole number of cows.

What was the neighbor's solution? 

Here is that solution! 

Source

Neighbor's Solution on Old Farmer's Will

What was the problem with the will?

Neighbor went & brought a extra cow. Now there were 18 cows. The oldest son got 
1/2 x 18 = 9 cows, middle son got 1/3 x 18 = 6 cows and youngest son got 1/9 x 18 = 2 cows. In this way, total 9 + 6 + 2 = 17 cows distributed among 3 sons but left with 1 cow which neighbor took with himself. 

Do Not Tap on the Glass

A man works at an aquarium. Every day he spends a large chunk of his time trying to stop people from tapping on the glass at the shark tank. Finally, fed up with it, he comes up with a solution. The solution works perfectly, the next day no one taps on the glass. However, he is fired for it. 

What was the man’s solution?

Read here that solution! 

Source 

'Reward' For 'Creative' Solution

But what made to use some creativity? 

The man did just a little trick. He painted a crack on the glass. Now people visiting stopped tapping on the glass with fear that it would break. But in that attempt man made the aquarium to look terrible & unsafe! Hence fired!

 

Motivated Man in Depression

A man in depression decided to commit suicide. He started walking along a railway track when he spotted an express train speeding towards him. Suddenly he changes his mind & decides not to suicide. To avoid it, he jumped off the track, but before he jumped he ran ten feet towards the train. 

Why?

Why did he do so? Click here to know! 

Logical Move By Depressed Man

Why his life was in danger? 

Since the man was firm on his decision of ending his life he started to walk on a railway bridge. By doing that, he made sure no body would save him & even if he falls in river, the death is sure as he didn't know swimming. After crossing the bridge more than half way, he changed his mind. Now by running back wouldn't be feasible option as train was speeding towards him at greater speed. Only way to save life was cross the bridge before train hits him.

Hence he ran 10 feet towards the train & jumped off the track thereby to save his life.

Corrupt Servant Stealing The Gold

You have 10 bags of gold coins.You have appointed 1 servant to carry each bag. One of your servants who were responsible for transport of the money wanted to trick you. He took one of the bags and filed away one gram of gold from each coin. One coin normally weighs 10 grams.

Can you figure out in one scaling which bag contains lighter coins? Which servant should be fired? – using digital scales (shows the exact weight of an item)?

Trick to get down the culprit! 

Source

Master Plan for Gold Stealing Servant

Where story begins? 

We need to take coins from each of 10 bags to test.

Number all the servants & their respective bags from 1-10. Now take 1 coin from first bag, 2 coins from second bag, 3 coins from third bag & so on to 10 coins from tenth bag. Put all those on digital scale. In normal case, with no corruption it would weigh in the case as 55 x 10 = 550 gm. But coins from one of the bag weighs 1 gm less. So if it weighs 1 gm less then 1st coin that came from first bag is filed one & hence first servant is corrupt. If it weighs 548 gm then coins from second bag are reduced ones & hence second servant is corrupt.

Depending on how many gm less than 550 gm it weighs the bag with manipulated coins can be identified & respective servant can be fired.  

Fair Distribution of Assets

A man is the owner of a winery who recently passed away. In his will, he left 21 barrels (seven of which are filled with wine, seven of which are half full, and seven of which are empty) to his three sons. However, the wine and barrels must be split, so that each son has the same number of full barrels, the same number of half-full barrels, and the same number of empty barrels. 

Note that there are no measuring devices handy. How can the barrels and wine be evenly divided?

This is how it can be done?

Source

Fairly Distributed Assets

What was dilemma? 

We have to make number of each barrel type divisible by 3.

1.Two half-full barrels are poured into 1 empty barrel. So we left with 8 full, 5 half-empty & 8 empty barrels.

2.Again transfer two half-fulled barrels into 1 empty. Now we have 9 full, 3 half-empty & 9 empty barrels.

Now it can be evenly divided as 3 full, 1 half-empty & 3 empty barrels (hence wine) among 3 sons. 

Mislabeled Boxes of Marbles

Three closed boxes have either white marbles, black marbles or both and they are labeled white, black and both. However, you’re told that each of the labels are wrong. You may reach into one of the boxes and pull out only one marble. Which box should you remove a marble from to determine the contents of all three boxes? 

 Mislabeled !

Find here how it can be done!  

Source 

Correcting labels of Marble Boxes

How they were mislabeled? 

We should pick marble from a box labeled as 'Both'. Since it is incorrectly labeled it must content either all black or all white marbles.  After we know what it contains, we can identify the other 2 boxes by the process of elimination as below.

1. If the marble picked is BLACK, then obviously it is the box of black marbles. Two remaining boxes labelled as WHITE & BLACK are left. Since all labels are wrongly tagged the box tagged with WHITE must not be the box with all white marbles. It must contain both colors of marbles. And the one tagged as a BLACK must contain all white marbles.
 
Look at the figures below. In circle, labels are written & outside are the contents of the boxes. Arrows represents how the contents are rotated to make labels of the boxes wrong.

2. If the marble is WHITE, then obviously it is the box of white marbles. Two remaining boxes labelled as WHITE & BLACK are left. Since all labels are wrongly tagged the box tagged with BLACK must not be the box with all black marbles. It must contain both colors of marbles. And the one tagged as a WHITE must contain all BLACK marbles.

'The Wisest Son'

Once an old called his three sons. He gave them equal money and ask them to buy something that can fill their living room entirely. He told them that he will give all his money and property to the son who is able to do this task as asked.

The first son buys sticks and tries to fill the room but he falls short of sticks. The second son buys straw but he also falls short of filling the room. The third son buys only two things and he is able to fill the room completely and thus earns all the property and money.

What did he buy?

Here is what did he buy! 

Source 

The Wisest Son Deserves Rewards

What was the condition? 

The third son was very wise to choose his option (like you did while choosing to read this blog). There are two/three possible things that he might have bought.

1. He bought a candle & match box. Just ignited that candle in room & light filled in entire room.

2. He just bought perfume & sprayed in room. The smell of perfume occupied the entire room. 

 

 

Terrible Dream Leading To Death - Puzzle

Two brothers were watching a horror film on video late one night. One brother dozed off and dreamed that he was being chased by the crazy man from the movie, who was trying to kill him. In the dream, he hid in a cupboard. There was no sound except his heart pounding, and he had no idea where his crazed captor was. He was terrified! At that moment, the video finished, and his brother put his hand on the shoulder of his sleeping sibling to wake him. The shock at that tense moment was enough that the sleeping brother suffered a massive heart attack and died instantly. True or false?

Know here reality! 

Source 

Script Writer of Terrible Dream Story!

What was the incident? 

Just a little lateral thinking required to crack that puzzle.

If that brother was died immediately after his brother put hand on his shoulder then how who wrote about his dream in detail? Hence the incident of death due to massive heart attack is totally fake one! Only possible way to know anybody about died brother's dream is the died brother himself coming in his dream & tell the entire story of what happened on that night. 

What is The Password Trick?

A young man was standing outside a bar thinking to enter into bar. But there was a one problem. There was a guard standing in door asking password. Only those producing correct password allowed to enter. The young man stood there & started observing carefully. A man came to enter into bar. The guard said - twelve, the man answered six. Man allowed to enter. A lady came, guard said - six; she replied - three.The guard allowed her to enter.


Now thinking that he has cracked the trick of password generation the young man went to the guard. The guard said - ten, immediately the young man replied - five. But the guard denied his entry to the bar. Can you tell the reason? 

What's the password?

This was the trick! 

To Generate The Correct Password

Read here what happened? 

The trick for the password was different from that the young man was thinking. 

The password was the number of letters in the spelling of number that guard was saying.

TWELVE - SIX

SIX -THREE

Hence for TEN the password should have been THREE not half of TEN! 

This was the code language designed for the entry into the bar. 

The Man in The Photograph

A man is looking at a photograph of someone. His friend asks who it is.

The man replies, 'Brothers and sisters, I have none. But that man's father is my father's son'.
 
Who is in the photograph?

Know here the relationship between! 

Source 

Relation With The Man in Photograph

What was the incident? 

The person in photograph is the man's son. Since the man doesn't have any 

brothers or sisters, the statement my father's son is himself. A shortened 

version would be this man's father is myself; so he is the father of the man in

the picture.

Who Will Shout First?

4 criminals are caught and are to be punished. The Judge allows them to be freed if they can solve a puzzle. If they do not, they will be hung. They agreed.
The 4 criminals are lined up on some steps (shown in picture). They are all facing in the same direction. A wall separates the fourth man from the other three.
So to summarize :-

Man 1 can see men 2 and 3.

Man 2 can see man 3.

Man 3 can see none of the others.

Man 4 can see none of the others.

The criminals are wearing hats. They are told that there are two white hats and two black hats. The men initially don't know what color hat they are wearing. They are told to shout out the color of the hat that they are wearing as soon as they know for certain what color it is.

They are not allowed to turn round or move.

They are not allowed to talk to each other.

They are not allowed to take their hats off.

Who is the first person to shout out and why?

Who will shout first?

Know that person here! 

Source