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. 

 

"Who is telling the truth?"

King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always say the truth.

One day, 4 servants met :

The blue one says: “Altogether we have 28 legs”;

the green one says: “Altogether we have 27 legs”;

the yellow one says: “Altogether we have 26 legs”;

the red one says: “Altogether we have 25 legs”.

 
What is the color of the servant that says the truth?

"I'm telling the truth and my color is..." 

"Listen to me; I'm telling the truth!"

What others making statements?

Since, each of 4 servants telling different numbers then only 1 of them must be telling the truth & other must be lying.

In that case, all 3 must be having 7 legs i.e. 21 legs while the one who is making true statement must have either 6 or 8 legs. Therefore, there must be 21 + 6 = 27 or 21 + 8 = 29 legs altogether.

Nobody is saying that they altogether have 29 legs but the green octopus is saying that they altogether have 27 legs. Hence, the green octopus must be telling the truth.