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

A ww by nn combination lock is comprised of ww wheels, each with the nonegative integers that are less than 10, in base nn, (mod nn), evenly spaced. For simplicity, we will assume that the combination code is (0,0,,0)Nw(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