**Motivation.** My youngest son has a bike lock with dials, and he forgot the unlocking combination completely, *except* that he remembered that digit $0$ appeared somewhere in the combination. So it was my task to go through all the zillions (but, fortunately, finitely many) possible combinations. Which led to the following problem.

**Formal statement.** Let $n \geq 2$ be an integer, so we have $n = \{0, \ldots, n-1\}$. For any integer $k>1$ let $$V^0_k = \{x \in n^k: (\exists j\in k)x(j) = 0\}, $$ and let two distinct elements $a\neq b \in V^0_k$ form an edge iff there is $j\in k$ such that

- $a(i) = b(i)$ for all $i\in n\setminus\{j\}$, and
- $\{a(j), b(j)\} = \{x, x+1\}$ for some $x\in n-1$, or $\{a(j), b(j)\} = \{0, n-1\}$.

Denote the set of edges by $E_k$.

For what positive integers $k, n$ does the graph $(V^0_k, E_k)$ have a Hamiltonian path? And, if there is a Hamiltonian path, can also a Hamiltonian cycle be found? (The second question doesn't need to be answered for acceptance.)

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