Puzzle : The Case of Missing Servant

A king has 100 identical servants, each with a different rank between 1 and 100. At the end of each day, each servant comes into the king's quarters, one-by-one, in a random order, and announces his rank to let the king know that he is done working for the day. 

For example, servant 14 comes in and says "Servant 14, reporting in." 

One day, the king's aide comes in and tells the king that one of the servants is missing, though he isn't sure which one.

Before the other servants begin reporting in for the night, the king asks for a piece of paper to write on to help him figure out which servant is missing. 

Unfortunately, all that's available is a very small piece that can only hold one number at a time. The king is free to erase what he writes and write something new as many times as he likes, but he can only have one number written down at a time. 

The king's memory is bad and he won't be able to remember all the exact numbers as the servants report in, so he must use the paper to help him.

How can he use the paper such that once the final servant has reported in, he'll know exactly which servant is missing?

Mathematical Trick to know the missing servant! 

Solution : The Missing Servant in the Case

What was the case?

When the first servant comes in, the king should write his number on the small piece of paper. For every next servant that reports in, the king should add that servant's number to the current number written on the paper, and then write this new number on the paper while erasing old one.

Addition of numbers from 1 to 100 = 5050.

Hence, 

Missing Servant Number = 5050 -  Addition of ranks of 99 Servants.

So, depending on how far the addition of 99 servants' rank goes to near 5050, the king can easily deduce the rank of missing servant.

For example, if the addition that king has after 99 servants report in is 5000 then the servant having rank = 5050 - 5000 = 50 must be missing. 

 

The Royal Love Story!

You’re a knight in love with a princess. Unfortunately, the king knows you’re poor and disapproves of the match.

On the night of a great feast, the king calls you up before his men and presents a golden box. In it are two folded slips of paper. One, he announces, reads “Marriage,” the other “Death.” “Choose one,” he says.

Pretending to stir the fire, the princess manages to whisper that both slips say “Death.” But the king and his men are waiting, and you cannot escape now.

What should you do?

This is how you can escape! 

Getting Your Love - 'The Princess'

What was the biggest challenge in getting that?

All you need to do is take either slip & drop it into the fire.

Now you can say, "Sorry, I couldn't manage to read my fate. But we have option to read what it was. It must be the opposite of what other slip reads.".

And obviously, other slip has 'Death' is written so no one can object that one that dropped in fire must had 'Marriage'. Any claim saying other too had 'Death' would raise the question of credibility of the king directly; so no one would dare to claim that.

Forgotten Bank Account Number

Today, John has to transfer 50 euro to the bank account of a Dutch friend. He has written down the account number on a piece paper. But since he had forgotten to take out the paper from his trousers when he put them in the washing machine, one digit of the bank account number became unreadable. The note says: 3170?4847. 

The friend of John is climbing the Mount Everest at the moment, so it is impossible for John to call his friend. Suddenly he remembers that a for a valid Dutch bank account number it holds that the first digit times 9 + the second digit times 8 + the third digit times 7 + ...... + the ninth digit times 1 should be divisible by 11. John thinks for a moment and finds the correct number. 

What is it? 

That it is !

Recalling Forgotten Bank Account Number

What were the clues? 

Let x be that missing digit. Then the bank account number looks like, 3170x4847.

Now 9 x First Digit + 8 x Second Digit + 7 x Third Digit.......must be divisible by 11.

That is 27 + 8 + 49 + 0 + 5x + 16 + 24 + 8 + 7 = 139 + 5x must be divisible by 11. 

The only value of x as a digit the satisfies above is 3. With that, total sum 154 is divisible by 11.

Hence, the account number must be, 317034847. 

 

"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!