On a periodic soliton cellular automaton

We propose a box and ball system with a periodic boundary condition periodic box and ball system (pBBS). The time evolution rule of the pBBS is represented as a Boolean recurrence formula, an inverse ultradiscretization of which is shown to be equivalent to the algorithm of the calculus for the 2Nth root. The relations to the pBBS of the combinatorial R matrix of {U'}q(AN(1)) are also discussed.