1. The best jump for the multi-minute button (MM) would be 8 minutes. In that case, the largest number of clicks would be 13. The worst case would be 54 minutes, which would require 6 clicks of the MM button followed by 7 clicks of the 1-minute button.

2. The jump sizes for MM in this case would be a descending sequence:
11, 10, 9, 8, 7, 6, 5
The largest number of clicks would then be 10. Advancing the minute hand by 10 minutes would require 10 clicks on the minute button. Advancing by 20 minutes requires one click of MM plus 9 clicks of the minute button. The worst case remains the same no matter how many times the multi-minute button is pressed.

3. If you have two multi-minute buttons, MM1 and MM2, then you could have one advance by 35 minutes and the other by 7. The worst case is 7 clicks, which occurs, for example, at 33 minutes (two clicks on the 35-minute button, then three on the 7-minute button, then two on the 1-minute button).

My colleague Richard Cole suggested the solution to problem 2.

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