The Short Circuit

Mike Morgan's technical journal

Closed form solution to Josephus problem

My 9 year old son Alexander came up with a couple of closed form solutions to the Joesphus problem.  Really.  If this helps you, please leave a comment, he will enjoy it.

A discussion of his solution (conjectures) and an attempted proof by me of one of them can be found as the first link in the references.

His main conjecture is:

$J(n)=2(n-2^{\lfloor\log_2 n \rfloor})+1, \mbox{ for } {n} \ge {1}$

where

$floor(x)=\lfloor{x}\rfloor$

is the largest integer not greater than x.

His other conjecture is in ceiling form is:

$J(n)= {(n-(2^{\lceil\log_2{n}\rceil}-1-n)) \mbox{ mod }2^{\lceil\log_2{n}\rceil}},\mbox{ for } {n} \ge {2}$

where

$\mbox{ceiling}(x)=\lceil{x}\rceil$

is the smallest integer not less than x.

The idea of mathematical proof is a little beyond Alexander right now, but he is planning to write something up on his discovery without my help. He is presently working on two other papers, both of which are impressive for a 9 year old, but this solution was so impressive I decided to use it as an opportunity to co-author a "little-league" paper with him to illustrate the ideas and presentation of mathematical ideas in an academic format. Hopefully I haven't done more harm than good with my attempt at a proof (formal mathematical proofs are not something engineers routinely do, so suggestions from the more mathematically literate are welcome).

Incidentally, my wife also decided to start the Journal of Mathematics for Minnesota Youth (JMMY), as a little league for bright youth. Alexander hopes that his friends at his new school next year are able to contribute to it. Hopefully we can entice some math teachers to become editors of it.

References