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The infected squares puzzle is a classic; you can read about it here. I'm interested in the $k$-dimensional version, specifically in establishing an upper bound.

Consider an $n\times n\times\cdots n$ $k$-dimensional hypercube. Choose some integer $C$; let all cells whose coordinates sum to $C$ mod $n$ be "infected" cells. A cell becomes infected if it shares a hyperface (=codim 1) with $k$ infected cells. Will the whole hypercube become infected?

If so, the $k$-dimensional, codimension-1 neighborhood, threshold $k$ variation of the infected squares puzzle has an upper bound of $n^{k-1}$. (In fact, this is a lower bound as well, but I won't prove it here; it's the same strategy as in the original infected squares puzzle.)

Experiments with $k=2,3$ support this, though I'm having a hard time definitively proving that this works with $k=3$, and my visualization skills become strained in the $k=4$ case.

Akiva Weinberger
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  • Fun fact: the $3\times3\times3$ case has 7 different optimal solutions (with 9 initially infected cells) up to symmetry, according to a computer search done by ZenoRogue – Akiva Weinberger Dec 01 '22 at 04:14
  • (Also, this is unrelated to the question at hand, but the threshold 3 2D case is fascinating. I believe the optimal solution for infecting the whole square is $\frac13n^2+\frac23n+c$, where $0\le c\le2$, and $c=0$ iff $n$ is one less than a power of two.) – Akiva Weinberger Dec 01 '22 at 04:22
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    This is proven in Mathematical Mind Benders, by Peter Winkler. – Mike Earnest Dec 01 '22 at 05:19
  • I spent $10$ years playing with the $k$-dimensional generalization of the infamous "Nine dots problem", so I have to pass... since, it could become more than addictive. – Marco Ripà Dec 01 '22 at 05:20
  • @MikeEarnest Ah, wonderful! I'll see if I can find an online copy of the relevant bit. – Akiva Weinberger Dec 01 '22 at 05:31
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    @MikeEarnest Found it! https://imgur.com/a/tCJ5p5W Wow, that's a nice argument (though I think there's at least one typo: it should be $x_i>x^$ the first time and $x_i<x^$ the second time). And they found the same configuration that I did! – Akiva Weinberger Dec 01 '22 at 05:57
  • The link you provide at the beginning of your question doesn't work (at least on my computer). Why that ? – Jean Marie Dec 01 '22 at 07:26
  • @JeanMarie Dunno… Try Firefox? – Akiva Weinberger Dec 01 '22 at 15:30

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