Partition and sum is fast

TL;DR

The paper investigates the efficiency of a 'partition and sum' operation on numbers, showing that it rapidly reduces any number to a single digit in surprisingly few steps.

Contribution

It introduces and analyzes the 'partition and sum' operation, demonstrating its quick convergence to a single digit.

Findings

Few iterations are needed to reduce any number to a single digit.

The operation is computationally efficient for large numbers.

The process always converges to a single digit regardless of the initial number.

Abstract

We consider the following "partition and sum" operation on a natural number: Treating the number as a long string of digits insert several plus signs in between some of the digits and carry out the indicated sum. This results in a smaller number and repeated application can always reduce the number to a single digit. We show that surprisingly few iterations of this operation are needed to get down to a single digit.