Base 10 RATS Cycles and Arbitrarily Long Base 10 RATS Cycles

Abstract

John Conway invented a digital game called RATS [1], an acronym for Reverse, Add, Then Sort. A game of RATS produces a sequence of positive integers. The decimal digits of each positive integer in the sequence are all nonzero and in nondecreasing order. The first number in a RATS sequence is a positive integer whose decimal digits are all nonzero and in nondecreasing order. To produce the (i + 1)st number in a RATS sequence from the i number, for i ≥ 1, reverse the digits of the ith number, add this number to the ith number, delete the zero digits in the sum, and then sort the remaining digits of the sum in nondecreasing order. The resulting sequence of positive integers is the RATS sequence for the first number.