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### Hamiltonian Path through $n$-bit strings with maximum number of $0\mapsto 1$ transitions

Let $G_n$ be the complete graph whose vertices are the $2^n$ $n$-bit strings. Let $H_n$ denote the Hamiltonian path through $G_n$ that uses the maximum number of edges that correspond to a single bit ...

**1**

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**1**answer

107 views

### How many non-overlapping k-hop neighborhoods can be uniquely colored on an $N$-dimensional hypercube?

Imagine I have a $N$-dimensional hypercube. My aim is to distinctly color as many non-overlapping $k$-hop neighborhoods as possible (i.e. sets of vertices connected by a Manhattan distance of at most ...

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843 views

### Sprague-Grundy sequence for the ruler game

Consider the game "Ruler", which is defined as follows. We start with finitely many coins in a line. A move in this game consists of turning over any number of coins, but they must be consecutive, ...