Math puzzle reference request

Question

A long time ago I was given the following math puzzle by a professor of mine:

First step: Take four numbers $a,b,c,d\in\mathbb{N}$.

Second step: Replace the 4-tuple $(a,b,c,d)$ by $(|a-b|,|b-c|,|c-d|,|a-d|)$.

Third step: Repeat step two with the new 4-tuple.

It is not hard to prove that eventually you reach $(0,0,0,0)$. My question is the following: Does anyone know the story behind this puzzle (i.e., what it's called, possible generalizations that have been explored, etc.)?

Answer 1

This is known as the "Four Number Game" or "Ducci's four-number problem" and appears to have been studied. See, for example:

• Benedict Freedman's 1948 paper: https://arxiv.org/abs/1109.0051 (actually from 1948, despite the URL). As an example of a generalization, if you start with an $n$-tuple, you're only guaranteed to get all zeroes if $n$ is a power of two.
• these notes by Daniel Shapiro, which include a bibliography: https://people.math.osu.edu/shapiro.6/4NumbersGame.pdf
• Wikipedia: https://en.wikipedia.org/wiki/Ducci_sequence