Happy Numbers, Happy Functions, and Their Variations: A Survey

Abstract

The happy function maps a positive integer (expressed in base 10) to the sum of the squares of its digits. A happy number is a positive integer that, under repeated applications of the happy function, maps to 1. The iterative behaviors of the happy function and its many variations have been the subject of study for over 75 years. In this survey, we discuss the arc of this research, explaining the ideas behind the methods and their general evolution. We explore and organize the wide range of variations, providing full proofs of the key results.