Inserting Plus Signs and Adding

Abstract

For a fixed base b, we consider the following operation called “insert and add” on a natural number: Write the number as a string of digits and between some of the digits insert a plus sign; finally, carry out the indicated sum. If at least one plus sign is inserted, this results in a smaller natural number and so repeated application can always reduce the number to a single digit. We show that for any base b, surprisingly few applications of this operation are needed to get down to a single digit.