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 $C$ mo...

Formally, how would I obtain that differentiability at $\vec a$ implies $\lim\limits_{t\to 0}\frac{f(\vec a+t\vec v)-f(\vec a)-f'(\vec a)t\vec v}{|t|}=0$ if we define differentiability as $f:\Bbb R^n \to \Bbb R^m$ is differentiable at $\vec a$ if and only if there exists a linear map $f'(\vec a):...

Theorem: Let $M$ be an $(R,S)$ bimodule, $N$ be a left $S$ module. Then $M\otimes_S N$ can be made a left $R$ module via the following action of $R$ on $M\otimes_S N$: $(r,\sum_{i=1}^tm_i\otimes_S n_i):= \sum_{i=1}^t(rm_i\otimes_S n_i)$. Here's a proof of this theorem that's along the lines of th...

There's a famous story about an exercise from Lang's Algebra that says something along the lines of "pick up a homological algebra book and prove all of the theorems yourself". I cannot find it in the third revised edition, and I'm wondering if it's still in the third revised edition, if it's o...

3D infected cubes puzzle with threshold $4$: On an $n\times n\times n$ cube, some cells are infected; if a cell shares a face with $4$ infected squares, it becomes infected. What's the minimum number of initially infected cells required to infect the whole cube? The two-dimensional, threshold 2...