The CryptArithmetic Problem's Solution

What was the problem?

Let's first recall the given equation.

  BASE +
  BALL
---------
GAMES
----------

We are assuming repeating the numbers are not allowed. 

Let's first take last 2 digits operation into consideration i.e. SE + LL = ES or 1ES (carry in 2 digit operation can't exceed 1). For a moment, let's assume no carry generated.

10S + E + 10L + L = 10E + S .....(1)

9 (E - S) = 11L

To satisfy this equation L must be 9 and (E - S) must be equal to 11. But difference between 2 digits can't exceed 9. Hence, SE + LL must have generated carry.So rewriting (1),

10S + E + 10L + L = 100 + 10E + S

9 (E - S) + 100 = 11L

Now if [9 (E - S)] exceeds 99 then L must be greater than 9. But L must be digit from 0 to 9. Hence, [9 (E - S)] must be negative bringing down LHS below 100. Only value of E - S to satisfy the given condition is -5 with L = 5. Or we can say, S - E = 5.

Now, possible pairs for SE are (9,4), (8,3), (7,2), (6,1), (5,0). Out of these only (8,3) is pair that makes equation SE + LL = SE + 55 = 1ES i.e. 83 + 55 = 138. Hence, S = 8 and E = 3.

Replacing letters with numbers that we have got so far.

    1---------
  BA83 +
  BA55
---------
GAM38
----------

Now, M = 2A + 1. Hence, M must be odd number that could be any one among 1,7,9 (since 3 and 5 already used for E and L respectively).

If M = 1, then A = 0 and B must be 5. But L = 5 hence M can't be 1

If M = 7, then A = 3 or A = 8. If A = 3 then B = 1.5 and that's not valid digit. And if A = 8 then it generates carry 1 and 2B + 1 = 8 again leaves B = 3.5 - not a perfect digit.

If M = 9, then A = 4 (A = 9 not possible as M = 9) and B must be 7 with carry G = 1.Hence for first 2 digits we have 74 + 74 + 1 = 149.

Finally, rewriting the entire equation with numbers replacing digits as -

    1
---------
  7483 +
  7455
---------
14938
----------

So numbers for letters are S = 8, E = 3, L = 5, A = 4, B = 7, M = 9 and G = 1.   
       

Black or White Dot on Forehead

There are five men, let's say Tarun, Harish, Lavesh, Manoj and Manish. All of them have a dot mark in their forehead. They can't see the dot on their own forehead, but can see the ones on others. The owner of WHITE dot is an honest person and will never lie, while the owner of BLACK dot always tell the lie.

This are the statement from Tarun, Harish, Lavesh, and Manish:

Tarun: 'I see 3 whites and 1 black'

Harish: 'I see 4 black'

Lavesh: 'I see 3 black and 1 white'

Manish: 'I see 4 white'

What color is the dot on each Tarun, Harish, Lavesh, Manoj, Manish forehead?

Know color of dot on each forehead!

Source 

Men with White Dots on Foreheads

Read this story first!

Let's take a look at who said what. 

Tarun: 'I see 3 whites and 1 black'
Harish: 'I see 4 black'
Lavesh: 'I see 3 black and 1 white'
Manish: 'I see 4 white'
 
Harish and Manish made very contracting statements; so are statement of Tarun and Lavesh. That means, only one of them is telling the truth. Hence, only one of 4 have WHITE dot on his forehead. Till now we don't have idea of what Manoj had on his forehead. Even if he has WHITE dot then there can be maximum 2 WHITE dots among those 5.    

Truth of Tarun and Manish:

Tarun and Manish must be lying as they are saying that they have seen 3 and 4 WHITE dots respectively. According to our first conclusion, there can be maximum 2 WHITE dots possible.

Truth of Harish:

Next if we assume Harish has WHITE dot and telling the truth then all other must be lying including Manoj and Lavesh. As per Lavesh, he had seen 3 BLACK and 1 WHITE dot. Now he must have seen BLACK dots on foreheads of Manoj, Tarun, Manish and WHITE dot on forehead of Harish as assumed. That mean he is telling truth though he has BLACK dot. But the man with BLACK is always lying, hence this case is also INVALID. 

Truth of Lavesh:

So the only person left is Lavesh and must be telling the TRUTH with WHITE dot on his forehead. Hence, as his statement is suggesting, the Manoj must have WHITE dot and other three Tarun, Lavesh, Harish have BLACK dots (we already concluded these 3 are lying). 

A Car on a Fragile Bridge

A car is crossing a 20km long bridge. The bridge can support at most 1500kg of weight over it. If somehow, the weight on the bridge becomes more than that, it will break.
Now, the weight of the car is exactly 1500kg. At the midway, a bird comes and sits on the roof of the car. This bird weighs exactly 200 gram. 

 
Can you tell if the bridge breaks at this point or not? 

Read what will happen next? 

Source 

 

Bridge Under Load

What was the situation?

At first look, the first impression would be that the bridge will break certainly. But if you wait for a while before concluding anything you will get the right answer.

The bridge will not break in the case! It's 20 km long bridge & now it's in the middle of bridge after traveling 10 km. By now, it must have used half of the fuel that was in the tank initially at the start. This amount of fuel must be weighing more than 200 gm. Hence even a bird sits on the car there are hardly any chances the total weight on the bridge goes beyond 1500 kg! Hence, no chance of breaking of it.

 

Mixture of Coffee and Tea

Suppose we lay down two cups in front of you. One of the cups is filled with tea and the other one with coffee. Now we ask you to take a spoonful of tea and mix it with the coffee. 

At this moment, the coffee cup has a mixture of tea and coffee. You have to take that mixture (spoonful) and add it back to the tea. 

Can you now tell if the cup of coffee has more tea or the cup of tea has more coffee ?

Skip to the answer! 

Source 

Component Levels in Mixture

How mixture made?

After adding 1 spoon of tea into coffee, the levels of liquids in both cups must be unequal. Whatever now tea cup is missing is now in cup of coffee & mixed with coffee. The content of tea in the cup of coffee is certainly more.

Now after taking spoonful of the mixture back to tea cup the levels of the liquid in both cups would be same. Hence, whatever the cup of tea is missing is replaced by coffee. That missing tea content is now in the cup of coffee where it has replaced some of coffee content! 

Suppose there are 1000 molecules in each cup i.e. of tea & coffee. Let's assume 100 molecules of tea are mixed to coffee using spoon. Now, coffee cup will have 1100 molecules and tea will have 900 molecules. Obviously, right now the cup of coffee contains more tea (100 molecules) that coffee in cup of tea (0 molecules)!

Now while taking 100 molecules back from mixture having 1100 molecules, suppose 70 molecules of coffee & 30 of tea are taken. That means, exactly 100 - 30 = 70 molecules of tea left in mixture. That 70 + 30 molecules mixture is poured into cup of tea. That is exact 70 molecules of coffee mixed in tea.

What does it mean? 70 molecules of coffee have displaced 70 molecules of tea into cup of coffee maintaining level of both the liquids. 

We can say other way as well. 30 molecules of tea displaced 30 molecules of coffee into cup of tea while maintaining levels of both the liquids same. 

So the answer is both have same level of contents mixed.

 

An Insepection by The Superintendent

One day, a class teacher was told that the school superintendent will be visiting her class on the next day. The superintendent can ask questions from anywhere and it can be easy as well as difficult. The teacher will have the liberty to choose any pupil for answering the question.

Now she is determined that the impression that is cast upon the superintendent after the inspection should be great. How will she instruct the students so that she maximizes the chances of receiving a correct answer for each question? Also, she must create the best impression. How will she do it? 

This is what she should do! 

To Impress Superintendent

What was the resolution of teacher? 

Now what should teacher do here is to devise the 'sign' language to communicate with students. Also she needs to make sure that the superintendent won't have any doubt while questioning students.

She should ask all the students to raise hands for every question that is being asked by superintendent. However, those who know correct answers should raise right hand & rest of all should raise left hand. This way she would be able to know the students who knows the correct answer & choose any of them to answer the question.

All raised hands to each question would definitely leave great impression on the superintendent.

Note : We are assuming superintendent not smart enough to notice that students raising different hands for different questions.

Generous Devotee

A devotee visits 9 temples when he visits India. All these nine temples have one thing in common - there are 100 steps in every temple. The devotee puts Re.1 coin after climbing up every step. He does the same while climbing down every step. At each temple, the devotee offers half of his money from his pocket to god. In this way, his pocket becomes empty after his visit to 9th temple.

Can you calculate the total amount he had initially ? 

Click here to know exact amount! 

Donations By The Devotee

Why to calculate those?

Using algebraic equations in the case can make things complicated unnecessarily. Hence, we would start from backward. Before putting 100 coins on steps while climbing down 9th temple devotee must had 100 coins. That means he had 200 coins when he climbed up the 9th temple half of which i.e. 100 he offered to that temple & 100 put on the 100 steps of 9th temple. Moreover, he must have placed 100 coins while climbing up 9th temple. So before visit to 9th temple he must had, (100 x 2) + 100 = 300 coins.

Same way, finding the amount he had before visit to each temple like below.

Before eight temple: (300+100)*2 + 100 = 900
Before seventh temple: (900+100)*2 + 100 = 2100
Before Sixth temple: (2100+100)*2 + 100 = 4300
Before fifth temple: (4300+100)*2 + 100 = 8900
Before fourth temple: (8900+100)*2 + 100 = 18100
Before third temple: (18100+100)*2 + 100 = 36,500
Before second temple: (36500+100)*2 + 100 = 73300
Before first temple: (73300+100)*2 + 100 = 146900

 
To conclude,  he had Rs. 146900 initially.  

 

Cars Across the Desert

A military car carrying an important letter must cross a desert. 

There is no petrol station on the desert and the car has space only for petrol that lasts to the middle of the desert.

There are also other cars that can transfer their petrol into one another.

How can the letter be delivered?

This is how letter can be delivered!

Source 

Delivering Letter Across The Desert

What was the task?

We need 4 such cars to deliver the letter across the desert successfully.

Let's divide the entire route into 6 parts. That means the distance that car can travel (half the total path in desert) is divided into 3 parts. To travel each part car requires 1/3rd of it's petrol in the tank.

1. At first 1/6th of total path, all cars are 2/3rd full. Now 2/3rd of the petrol from 1 car can be used to fill 1/3rd of tanks in other 2 cars (1/3 + 1/3 = 2/3). This way, we would have 2 cars full while 1 car 2/3rd full. We are leaving behind the empty car, taking 3 cars forward.

Stage 1

2. At next 1/6th of the distance, 2 full cars will use 1/3rd of their petrol hence would be 2/3rd full. And the car that was 2/3rd at previous stage would be not 1/3rd full. At this stage, the petrol from car that is 1/3rd full can be used to fill tank of 1 car completely. So we are leaving behind one another empty car here & taking fully filled car & 2/3rd filled car for next stage.

Stage 2

3. For next 1/6th of the total distance, the car that was fully filled would have 2/3rd petrol. And the car which was 2/3rd at previous stage would be now 1/3rd filled. The petrol of this car can be used to fill the tank of the first car. Now we have 1 car fully filled while other one is empty. So we can leave behind the empty car & use fully filled car for the rest half of the journey. Remember, a car which tank is full can travel half the total path.

Stage 3

 

Who Works Where?

Alex, Betty, Carol, Dan, Earl, Fay, George and Harry are eight employees of an organization
They work in three departments: Personnel, Administration and Marketing with not more than three of them in any department.

Each of them has a different choice of sports from Football, Cricket, Volleyball, Badminton, Lawn Tennis, Basketball, Hockey and Table Tennis not necessarily in the same order.

1.Dan works in Administration and does not like either Football or Cricket.

2.Fay works in Personnel with only Alex who likes Table Tennis.

3.Earl and Harry do not work in the same department as Dan.

4.Carol likes Hockey and does not work in Marketing.

5.George does not work in Administration and does not like either Cricket or Badminton.

6.One of those who work in Administration likes Football.

7.The one who likes Volleyball works in Personnel.

8.None of those who work in Administration likes either Badminton or Lawn Tennis.

9.Harry does not like Cricket.

 
Who are the employees who work in the Administration Department?

In which Department does Earl work?

Who is the fan of each sports?

Click here for the complete picture. 

Source 

Employees of Each Department

What was the data given? 

Let's make a table where columns represent the sport & row represents the employee.There are 3 tables 1 for each department. To make table shorter we will use the initials only of sports' & employees' names as below.

Table

A - Alex, B - Betty, C - Carol, D - Dan, E - Earl, F - Fay, G - George, H - Harry. 

F - Football, C - Cricket, V - Volleyball, Bd - Badminton, LT - Lawn Tennis, Bs - Basketball,
H - Hockey, TT - Table Tennis  

Now taking clues one by one into consideration.

1. Dan works in Administration and does not like either Football or Cricket.

Table 1

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

2. Fay works in Personnel with only Alex who likes Table Tennis.

This indicates that Alex is working in Personnel department & likes Table Tennis. Fay working in same department may like any other sports than Table Tennis. No body other working in this department.

Table 2

--------------------------------------------------------------------------------------------
3. Earl and Harry do not work in the same department as Dan.

Hence they must be working in Marketing department!

Table 3

---------------------------------------------------------------------------------------------
4. Carol likes Hockey and does not work in Marketing.

That's why his department must be Administration.

Table 4

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5. George does not work in Administration and does not like either Cricket or Badminton.

His department must be Marketing & he might be liking Football or Volleyball or Lawn Tennis or Basketball.

Table 5

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The Seven Rings

You arrive at a hotel and have 3 sets of golden rings. The first set of rings has 4 rings, the second set has 2 rings and the third only has one ring. You cannot take these sets of rings apart, exchange them for a different form of currency, and the hotel clerk has no change. You want to stay at the hotel for 7 nights, and you have to pay one gold ring for each night that you stay. You cannot pay in advance, or all at once at the end of your stay.

 How do you pay for your 7 nights at the hotel?

This is how should you pay! 

Source 

Paying Rings At Hotel

What was the condition? 

You can pay 7 rings in 7 days in following sequence.

Day 1 : 

Give the only ring that is in first set. Paid 1 ring.

Day 2 : 

Take back ring given on Day 1 & give second set of rings having 2 rings. Paid 2 rings

Day 3 :

Give 1 ring back again. Total rings paid = 2 + 1 = 3

Mathematical Coincidence

Messi entered a candy shop and spent half of the money in his pocket. When he came out he found that he had just as many paise as he had rupees when he went in and also half a many rupees as he had paise when he went in. How much money did he have on him when he entered? (1 Rupee = 100 Paise just like 1 Dollar = 100 Cents)

Know the amount that Messy had initially!!

Source 

Money For Mathematical Coincidence

What was the coincidence?

Let X be the rupees & Y be the paise that Messy initially had in his pocket. That means he had 100X + Y paise initially. In shop he paid half of the amount he had i.e. (100X + Y)/2. 

When he came out of the shop he had Y/2 rupees & X paise i.e. 100(Y/2) + X paise.

Equating both,

(100X + Y)/2 = 100(Y/2) + X

Multiply by 2,

100X + Y = 100Y + 2X

98X = 99Y


Hence, X = 99 & Y = 98
 
Putting these values in 100X + Y = 9998. This is the amount in paise. Converting in rupees gives, 9998/100 = 99.98

 
To conclude, Messi had Rs.99.98 initially in his pocket.


 

Correlation Of Playback Speed & Duration

Recently youtube added speed control on video playback on mobile app. Previously it was only for desktop browsers. Now, if I increase the speed to 1.25 then how much time I would save while viewing particular video? Would it take 25% less time than original video? 

Interpreting it like that way is totally wrong way. Without going too much into technical terms, let's name 'content' for the whatever video has for it's entire duration. One thing is sure with playback speed of increased the duration for which we would view is reduced. The (oversimplified) formula in this case should be,

The Tuesday Birthday Problem

I ask people at random if they have two children and also if one is a boy born on a Tuesday. After a long search I finally find someone who answers yes. What is the probability that this person has two boys? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day.

Tip: Don't conclude too early. 

Click here to know the correct answer! 

Finding The Correct Probability

How tricky it was?

If you think that the probability is 1/2 after reading that the couple has equal chance of having child of either sex then you are in wrong direction.

Take a look at the table below.

There are 27 possible combinations when boy is born on Tuesday. Out of which there are only 13 possible combinations where either boy (first or second) is born on Tuesday. 

Hence the probability that the person having at least 1 boy off his 2 boys born on Tuesday is 13/27.

Distinguish The Fake Coin

You have twelve coins. You know that one is fake. The only thing that distinguishes the fake coin from the real coins is that its weight is imperceptibly different. You have a perfectly balanced scale. The scale only tells you which side weighs more than the other side.

What is the smallest number of times you must use the scale in order to always find the fake coin?
 
Use only the twelve coins themselves and no others, no other weights, no cutting coins, no pencil marks on the scale. etc.

These are modern coins, so the fake coin is not necessarily lighter.

Presume the worst case scenario and don't hope that you will pick the right coin on the first attempt.

Process to identify the fake one! 

Source 

Process To Identify Fake Coin

What was the task given? 

If we knew, the fake coin is lighter or heavier than original one then the process would have been pretty simple like this! But we don't know.

Let's number the coins from 1 to 12. We'll make 3 groups of these coins as 1,2,3,4 in one group, 5,6,7,8 in other group and 9,10,11,12 in one more group.

First of all weigh 1,2,3,4 against 5,6,7,8.

CASE 1 : 1,2,3,4 = 5,6,7,8

 That means coin among 9,10,11,12 is fake one. So weigh 9,10 against 11,8.

   CASE 1.1 : If 9,10 = 11,8 then 12 is fake coin.

   CASE 1.2 : If 9,10 > 11,8 then either 9 or 10 is heavier (hence fake) or 11 is lighter (hence fake). Weigh 9 against 10. If they balance then 11 is fake one. If they don't then heavier of 9 & 10 is fake. 

   CASE 1.3 :  If 9,10 < 11,8 then either 9 or 10 is lighter (hence fake) or 11 is heavier (hence fake). Weigh 9 against 10. If they balance then 11 is fake one. If they don't then lighter of 9 & 10 is fake.

The Coconut Problem

Ten people land on a deserted island. There they find lots of coconuts and a monkeys. During their first day they gather coconuts and put them all in a community pile. After working all day they decide to sleep and divide them into ten equal piles the next morning.

That night one castaway wakes up hungry and decides to take his share early. After dividing up the coconuts he finds he is one coconut short of ten equal piles. He also notices the monkey holding one more coconut. So he tries to take the monkey's coconut to have a total evenly divisible by 10. However when he tries to take it the monkey conks him on the head with it and kills him.

Later another castaway wakes up hungry and decides to take his share early. On the way to the coconuts he finds the body of the first castaway, which pleases him because he will now be entitled to 1/9 of the total pile. After dividing them up into nine piles he is again one coconut short and tries to take the monkey's slightly bloodied coconut. The monkey conks the second man on the head and kills him.

One by one each of the remaining castaways goes through the same process, until the 10th person to wake up gets the entire pile for himself. What is the smallest number of possible coconuts in the pile, not counting the monkeys?

Here is that smallest number! 

Source 

Number Of Coconuts In The Pile

What was the problem? 

Absolutely no need to overthink on the extra details given there. Just for a moment, we assume the number of coconuts in the community pile is divisible by 10,9,8,7,6,5,4,3,2,1.

Such a number in mathematics is called as LCM. And LCM in this case is 2520. Since each time 1 coconut was falling short of equal distribution there must be 2519 coconut in the pile initially. Let's verify the fact for all 10 distributions tried by 10 people.Each time monkey kills 1 person & number of persons among which coconuts to be distributed decreases by 1 each time.

Wise Men In Survival Game

A stark raving mad king tells his 100 wisest men he is about to line them up and that he will place either a red or blue hat on each of their heads.

Once lined up, they must not communicate among themselves. Nor may they attempt to look behind them or remove their own hat.The king tells the wise men that they will be able to see all the hats in front of them. They will not be able to see the color of their own hat or the hats behind them, although they will be able to hear the answers from all those behind them.

The king will then start with the wise man in the back and ask "what color is your hat?" The wise man will only be allowed to answer "red" or "blue," nothing more. If the answer is incorrect then the wise man will be silently killed. If the answer is correct then the wise man may live but must remain absolutely silent.The king will then move on to the next wise man and repeat the question.
 
The king makes it clear that if anyone breaks the rules then all the wise men will die, then allows the wise men to consult before lining them up. The king listens in while the wise men consult each other to make sure they don't devise a plan to cheat. To communicate anything more than their guess of red or blue by coughing or shuffling would be breaking the rules.

What is the maximum number of men they can be guaranteed to save?

Almost all can survive! Click here to know! 

Source