Equitable Candy Sharing

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.