Math Geniuses. I call upon you.?

Question

The King sits at his round table with his knights. He needs to choose a husband for his daughter. He kills every other knight until there is one left. The first lives, the second dies, the third lives, and so on. Find the formula to describe where you should sit to live for any given number of knights.

Answer 1

S(n) = 2(n - 2^k) + 1

n = number of knight around the table

2^k ≤ n < 2^(k+1)

Answer 2

Interesting question.

THINKING!

*strokes beard*

I'm trying to think of an algorithm to survive.

obviously you don't want to be an even number.

Here's a start:

if N=2^x for integers x. then knight 1 lives

if N=2^x-1 for integers x, last knight lives.

if N=2^x+1 for integers x, knight 3, lives.

I'm starting to get an idea here.  I'll keep working on it in the morning.

In general:

N - 2^x = p, (N=knights, x= floor(log[2](N))

Place to sit = 2p+1

Answer 3

Not sure of the formula yet, but the sequence is:

1, 1, 3, 1, 3, 5, 1, 3, 5, 7, etc

ie odd numbers in the pattern:

1

13

135

1357

13579