A padlock has two wheels, each showing the digits to equally spaced. Together they show a two-digit code. Trying the wrong code locks it forever, but a probe checks whether the code is close without locking the lock: it is close if either (i) the code is correct, or (ii) the correct code can be obtained by rotating one wheel to an adjacent digit (with adjacent to ).
For example, if the wheels show , the probe reports close for correct codes , , , and .
Find the minimum number of probe uses that guarantees discovering the correct code.
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Hamilton Mathematical Olympiad (2026)
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