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.