# Does this problem have a name? [Ducci Sequences]

Let $a_1, ... a_n$ be real numbers. Consider the operation which replaces these numbers with $|a_1 - a_2|, |a_2 - a_3|, ... |a_n - a_1|$, and iterate. Under the assumption that $a_i \in \mathbb{Z}$, the iteration is guaranteed to terminate with all of the numbers set to zero if and only if $n$ is a power of two. A friend of mine knows how to prove this, but wants to be able to reference a source where this problem (and/or its generalization to real numbers) is mentioned, and we can't figure out what search terms to use. Can anyone help us out?

(If someone can figure out a better title for this question, that would also be appreciated.)

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This is known as Ducci's problem. "Ducci map" or "Ducci sequence" as key words should let you search on most of the articles studying properties of the above map.

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Thanks! This is really helpful. –  Qiaochu Yuan Mar 8 '10 at 1:35