Dear NRICH Team,
I believe I have found a winning technique, if you go first:
You want to get to midnight (00:00). To force your opponent to get you to midnight, you need to land on 10:45 pm.The other player only has four options: 15 m (11 pm), 30m (11:15 pm), 45m (11:30 pm) or 60m (11:45 pm). If he land on any of these times, then you have the opportunity to go to 12 am by countering his move by making his and your time add up to midnight. Note that 10:45 is 75 minutes before midnight. This is because 15 + 60, 30 + 45 and all the other moves your opponent can make + the move you counter with add up to 75.
By similar reasoning, to get to 10:45 pm, you need to get 9:30 pm. Therefore, you need to get to 8:15 pm to get to 9:30 pm. Therefore you need to get to 7:00 pm from the initial time of 6:00 pm. Simple - just pick your first move as 60 m. Then match your opponent each time to get to the 'winning' times and you'll win!
(Also, if you go 2nd and your opponent doesn't land on 7:00 pm on the first move, make a move to land on 7:00 pm. Then apply the same tactics as above)