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For $n\in \mathbb{N}$ let $S_n$ denote the set of permutations (bijections) $\pi: \{0,\ldots,n-1\}\to \{0,\ldots,n-1\}$. A transposition swaps exactly $2$ elements and is often denoted by $(i \; k)$ i …

The following is inspired from the most recent riddle of the week of the German news magazine Der Spiegel. For any positive integer $n\in\mathbb{N}$, let $[n]$ denote the set $\{1,\ldots,n\}$. Let $E_ …

Fedor Petrov's answer to my more general question implies a positive answer to this question, as Banach spaces are complete metric spaces.

Let $G = (V,E)$ be a finite, connected, simple, undirected graph. By a roundtrip of $G$ we mean a map $r:\{0,\ldots,n\} \to V$ for some $n\in\mathbb{N}$ with the following properties: $r$ is surject …

Motivation. In their paper about the cryptographic scheme NORX, the authors use a fast approximation of + by bitwise operations (taking fewer CPU cycles than proper addition) using the formula $$a+b " …

In the following, we define infinite Hadamard matrices. Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are approximately orthogonal if $$\lim …

Let $f:\mathbb{N} \to \{0,1\}$ be the Ehrenfeucht-Mycielski sequence. The first few digits of the sequence are: $$010011010111000100001111\ldots$$ For any $k\in\mathbb{N}$ let $s(k) = \sum_{i=0}^k f(i …

Motivation. Recently I stayed at a hotel which had the curious custom to ask their $n$ parties (group of guests, most parties a married couple) which of the $n$ tables they wanted to take. Of course t …

Donald Knuth suggested a bitwise approximation for addition on the non-negative integers that is very fast on common processors: $(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$, where $a,b$ are g …