Reverse divisors are Ball numbers

Abstract

This study investigates the relationships between Ball magic numbers and reverse divi-sors. Specifically, we extend the set of numbers that are simultaneously reverse divisorsand Ball magic numbers. Webster and Williams [16] classify reverse divisors into threetypes: basic of the first and second types, and non-basic ones. The basic of the firsttype basics are numbers that begin with 10 and end with 89, with all intermediate digitsbeing 9, for example, 1099989. The basic of the second type is double that of a firsttype basic, such as 2199978. Non-basic reverse divisors are obtained by concatenat-ing magic reverse divisors. Costa and Santos [6] proved that a first type basic reversedivisor is a Ball’s magic number. This work demonstrated that a non-basic reverse di-visor, obtained by concatenating first-type basic reverse divisors, is also a Ball’s magicnumber.