Recently Ke Shi and I showed that \(\mathbb{Z}_n\) can be covered by \(O\left(\frac{n}{\ell}\log n\right)\) sets of the form \(\{ax\pmod n \mid 0\leq x \leq \ell \}\), where \(a\in \mathbb{Z}_n\). arxiv.org/abs/2207.05017

Turns out the algorithm is related to zero forcing sets. A fun concept. I found some nice slides on it. math.nsysu.edu.tw/~chlin/Publi

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For a \(n\times n\) grid, the lights out game can be considered as solving a linear system over \(n^2\) variables in mod \(2\). It takes \(O(n^6)\) time. Surprisingly, there is a \(O(n^3)\) time algorithm. chaoxuprime.com/posts/2019-01-

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