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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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up vote 11 down vote accepted

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

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