## Sunday, May 11, 2014

### Problem

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?

### Solution

When the first servant comes in, the king should write down his number. For each servant that reports in, the king should add that servant's number to the current number total. Once the final servant has reported in, the number on the paper should equal

1 + 2 + 3 + ... + 99 + 100) - Missing Servant's Number.

If they had all shown up it would have been 5050, since 1 + 2 + 3 + ... + 99 + 100 = 5050.

So to get the number of the missing servant just subtract the number that is on the paper from 5050.

5050 - Amount On Paper = Missing Servant's Number.