Game Of Death - Josephus Problem

There are 100 people standing in a circle in an order 1 to 100. No. 1 has a sword. He kills the next person (i.e. No. 2) and gives the sword to the next (i.e. No. 3). All people do the same until only 1 survives. Which number survives at the last?

Find who survived? 

Source 
 

The Man Surviving in Game Of Death

What was the problem? 

First let's make it very simple by who are surviving after each round.

Round 1 : 1,3,5,7,9,11,13,15..........87,89,91,93,97,99

Round 2 : 1,5,9,13,17.........89,93,97 

Round 3 : 1,9,17,25,33,41,49,57,65,73,81,89,97

Round 4 : 9,25,41,57,73,89

Round 5 : 9,41,73

Round 6 : 9, 73

At round 5, 9 kills 41 & passes sword to 73. So 73 kills 9 & survives.

Round 7 : 73

Now let's analyze  how this happens & trick to get answer at the quickest.

Just for a moment let's assume there were 16 standing in circle.

Now after each round survived people are,

Round 1 : 1,3,5,7,9,11,13,15

Round 2 : 1,5,9,13

Round 3 : 1,9

Round 4 : 1

After round 2, when 9 kills 13 & passes sword to 1.

And in round 4, 1 kills 9. So 1 forms pair with other in every round.

Now imagine there were 17 people in circle.