Equitable Candy Sharing

TL;DR

This paper analyzes a candy sharing process among children arranged in a circle, providing a necessary and sufficient condition for the system to reach an equitable distribution where each child has one candy, especially when total candies equal children.

Contribution

It introduces a precise condition under which the candy sharing process converges to an equitable distribution in the specific case where total candies equal children.

Findings

System reaches equitable distribution under specific initial conditions.

Necessary and sufficient condition for convergence is established.

Analysis focuses on the case where total candies equal children.

Abstract

Children, sitting in a circle, each have a nonnegative number of candies in front of them. A whistle is blown and each child with more than one candy passes one candy to the left and one to the right. The sharing process is repeated until a fixed state is attained, or the system enters a periodic cycle. This paper treats the case where the total number of candies equals the number of children. For a given initial distribution of candies, a necessary and sufficient condition is given for the system to ultimately attain the equitable distribution in which each child has one candy.