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! 

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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 

Master Plan By Wise Men

Why this master plan needed? 

99 can be guaranteed to save! How?

Even if the person behind calls out the color of the hat that next person is wearing both would be survived only if they are wearing same color of hat. 

So how 99 can be saved?

For a simplicity, let's assume there are only 10 wise men & (only) assume we are among them. Now, we need to make a master plan to survive from this game of death.

One of us need to agree to sacrifice his life to save 9 of us & this person would be the first one in line. He will be survived of he has good luck.

The first person in line should shout RED if he founds number of RED hats even otherwise he should shout BLUE. Now if he has good luck then the hat color of his own hat would match & he would be survived.

The clue given by the first person is very important. Right from second person everyone need to count number of RED hats in front of him. Additionally, the next person need to keep track of number of RED hats that people behind him are wearing.

The Greek Philosophers

One day three Greek philosophers settled under the shade of an olive tree, opened a bottle of Retsina, and began a lengthy discussion of the Fundamental Ontological Question: Why does anything exist?

After a while, they began to ramble. Then, one by one, they fell asleep.
 
While the men slept, three owls, one above each philosopher, completed their digestive process, dropped a present on each philosopher's forehead, the flew off with a noisy "hoot." Perhaps the hoot awakened the philosophers.

As soon as they looked at each other, all three began, simultaneously, to laugh.

Then, one of them abruptly stopped laughing. Why?

Interesting reason behind it! 

Source 

Theory Of The Smartest Philosopher

What's the story behind? 

The one who stopped laughing was the smartest one among! Read how he was the smartest.

We need to think from the smartest Philosopher's point of view. Let's name 2 other Philosophers as A & B.

Now here is what the smartest Philosopher would think.

"If I had nothing on my head then A & B must have been laughing after looking each other's head. And at least one of them is smart enough to realize that the other is laughing only after looking at him. That means A (or B) would have realized that the some thing is on his head too as B (or A) is laughing after looking him (not me if I had nothing on my head). Hence one of them would have stopped laughing. Since they are not stopping to laugh, I too must have something on my head."

Hence the smartest Philosopher stopped laughing after realizing that the fact.

  
A brilliant puzzle based on the similar logic is here!

Logic Problem: The Trainee Technician

A 120 wire cable has been laid firmly underground between two telephone exchanges located 10km apart.Unfortunately after the cable was laid it was discovered to be the wrong type, the problem is the individual wires are not labeled. There is no visual way of knowing which wire is which and thus connections at either end is not immediately possible.

You are a trainee technician and your boss has asked you to identify and label the wires at both ends without ripping it all up. You have no transport and only a battery and light bulb to test continuity. You do have tape and pen for labeling the wires.

What is the shortest distance in kilometers you will need to walk to correctly identify and label each wire?

Know here the efficient way! 

Source 

To Be A Skilled Technician

What was the task to test the skill? 

The shortest distance is 20 km! Surprised? Read further.

Let's name the two exchanges as a 1 & 2 respectively. Now at end 1, let's make a groups of wires having 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 number of wires. Now somebody might ask why not 15 groups having 8 wires in each. After reading the entire process here, we'll get the answer of it.So we 15 groups have total 1 + 2 + 3....+ 15 = 120 wires. Let's name these groups as A, B, C, D ...... O. That means group A has 1, B has 2, C has 3 wires & so on.

Now join together all the wires of the particular group. For example, 2 wires of group B should be joined together, 7 wires of G tied together & so on. The sole wire of A is left as it is.

We will take the battery & bulb to other end traveling 10 km. We will say a wire is paired with the other if the bulb gets illuminated if battery & bulb connected in between.

Now let's take any wire at the other end & find the number of wires that are pairing with that particular wire under test. We will group such wires & label with those exactly how we labeled at end 1.

For example, if we find 2 wires pairing with particular wire then that wire & 2 paired wires together to be grouped in 3 wires & labeled as C.The sole wire not getting paired with any will be labeled as A. And group with wire pairing with 7 other wires together should be labeled as H.

In this way, we will have the exact group structure that we have at end 1. By now, we have identified & labeled correctly wires in groups of 1,2,3,....15 wires at both ends.

Now, we are going to label each wire of group by it's group & count number. For example, the only wire in group A labeled as A1, 2 wires in B are labeled as B1,B2, wires in G are labeled as G1,G2,G3,G4,G5,G6,G7 and so on.

Now, at end 2 itself, what we are going to do is connecting first wire of each group to A1. Second wire of each group to be connected to B2, third of each to be connected to C3 and so on. (refer the diagram above, where labels of wires that are to be connected together are written in same color).

Magical Water Well & Pilgrim

In a small town, there are three temples in a row and a well in front of each temple. A pilgrim came to the town with certain number of flowers.

Before entering the first temple, he washed all the flowers he had with the water of well. To his surprise, flowers doubled. He offered few flowers to the God in the first temple and moved to the second temple. Here also, before entering the temple he washed the remaining flowers with the water of well. And again his flowers doubled. He offered few flowers to the God in second temple and moved to the third temple. Here also, his flowers doubled after washing them with water. He offered few flowers to the God in third temple.

There were no flowers left when pilgrim came out of third temple and he offered same number of flowers to the God in all three temples. What is the minimum number of flowers the pilgrim had initially? How many flower did he offer to each God?

Click here to know answers! 

Source 

Flowers Owned By Pilgrim

What are the questions? 

Let's assume that the pilgrim had X number of flowers initially. Suppose that he offered Y flowers to the each God. 

Before entering into the first temple, magical well doubled the flowers he had initially. That means he had 2X flowers before entering into first temple. After offering Y flowers, he had 2X - Y flowers.

Again after visiting first temple he washed flowers in magical well where number of flowers gets doubled. Now, he had 4X - 2Y.

Out of these 4X - 2Y, he offered Y flowers to God in second temple. So he had 4X - 2Y - Y = 4X - 3Y flowers after visiting second temple.

These 4X - 3Y doubled to 8X - 6Y after washing in magical water well. 

At third temple, he offered all the flowers he had which in turn equal to Y as we assumed.

8X - 6Y = Y

8X = 7Y

X/Y = 7/8

This is the ratio of the flowers that pilgrim had to the flowers he offered to each God.

In general, the pilgrim had 7N flowers initially and he offered 8N flowers to each God, where N = 1, 2, 3, 4,

Let's cross verify same with N = 1, meaning that the pilgrim had 7 flowers initially & offered 8 flowers to each God.

Before entering into first temple, the flowers doubled to 14. Out of which, 8 offered at first temple & left 6.

Again 6 doubled to 12 at magical well & 8 out of 12 offered to God at second temple leaving behind 4.

Those 4 again doubled to 8 by magical well & all 8 offered to God at third temple. 

Time Taken For The Journey

A RED ant is sitting on one side of a table (point X) and a BLACK ant is sitting on the opposite side of the table (point Y).

Now both of them decides to exchange their places and starts crawling. On the way, both of the ants meet and after that, it takes 20 seconds for the RED ant to move to point Y and it takes 5 seconds for the BLACK ant to reach point X.

Find out the total time taken by the RED and the BLACK ant to make the journey.

Here are steps to calculate! 

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