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Few days ago, a friend of mine gave me this problem :

Let $X \subset \mathbb{Z}$ and suppose that $a_1, a_2, \cdots, a_n$ are integers such that $X+a_1, \cdots, X+a_n$ is a partition of $\mathbb{Z}$. Prove that $X+p=X$ for some nonzero integer $p$.

I found a solution after sometime (my solution is by using indicator function $v_i = 1$ for $i \in X$ and $0$ otherwise).

But I'm more interested on the source of problem since I'm pretty sure I've seen this problem before, maybe in a blog post or mathematics competition. It's bugging me since I can't remember where did I read this problem before.

I've tried googling it, with no luck.

I'm more interested on the source, can someone tell me where is this problem from?

Ajat Adriansyah
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  • This problem is at least very similar to a basic result of Group Theory - that cosets partition a group (here, $\mathbb{Z}$ is a group). While I don't have source that investigates it without abstract algebra, Dummit and Foote's Abstract Algebra, 3rd Edition proves it as Proposition 4 in section 3.1

    $X\subset\mathbb{Z}$ with no restrictions on $X$ may make this analogy inexact, but it might be worth investigating anyway.

     – Mark Schultz-Wu May 23 '16 at 05:44
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    I'm afraid it's quite different mark. The coset requires them to be equivalence classes, i.e one of the $X+a_i$ which contains zero must be a subgroup.

    Here $X$ is not necessarily a subgroup of $\mathbb{Z}$, so $X+a_i$ is not necessarily a coset. For example let $X={ s \in \mathbb{Z} : s \equiv 0 , 2 \pmod 8 }$ , then $X$ is not a subgroup of $\mathbb{Z}$. While $X$, $X+3$, $X+4$, $X+7$ form a partition of $\mathbb{Z}$.

     – Ajat Adriansyah May 23 '16 at 05:51

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