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$\text{\Huge The Shortest Path of a Combination Lock}$

A $w$ by $n$ combination lock is comprised of $w$ wheels, each with the nonegative integers that are less than 10, in base $n$, (mod $n$), evenly spaced. For simplicity, we will assume that the combination code is $(0,0,\dots,0)\in\mathbb{N}^w$.

A click is rotation of one or more adjacent wheels so that the number(s) in the primary position of the wheel(s) that are being moved changes by one

Here we are interested in the shortest