Thank you to Mariah for sending in her strategy to Approaching Midnight.
The way to win 'Approaching Midnight' is to make the computer's final turn at 10:45, so no matter what they press (15, 30, 45 or 60 minutes) you will be able to reach midnight first.
This is because 10:45 is 1 hour and 15 minutes before 12:00. 1 hour and 15 minutes is the only possible time that can be reached in a round of your turn and the computer's, as 1 hours and 15 minutes equals 75 minutes. If the computer choses 60 minutes then you will chose 15 to equal 75, the same goes for 45 and 30, 30 and 45, 15 and 60.
The way to make sure the computer's final turn starts at 10:45 is using the same 75 minute pattern.
Starting first, you press 60 minutes, from then on, (so starting at the computer's first turn) every round must equal 75 minutes (60 minutes + 75 minutes x 4 = 360 minutes or 6 hours, the needed time).
Your next move, and every move after will depend on the computer. If the computer moves the clock forward 60 minutes, you move 15, in the same way you would on the final turn, as explained above.