From the instant shown to 3:00 exactly, how many times does the seconds hand pass over the minute hand, not counting the start and end instants?

The seconds hand turns 360° a minute and the minute hand 6°, so the seconds hand gains 354° a minute. In 60 min it gains 21 240° = 59 full turns. Both hands are together at 2:00:00 and again at 3:00:00 (the 59th meeting), so strictly between there are 59 − 1 = 58 passes. (59 counts the meeting at 3:00; 60 assumes one pass per minute.)