A number of pupils from Mortimer School in England sent in their solutions.
Here are their comments:

Alice and Oliver:
When your partner is doing 1/2  an hour you should do a 1 hour.

Aaron and Mustafa:
Get the clock on to half past ten and then try and make him use half an hour and you use 1 hour so it goes on to 12 o'clock.

What would happen if your partner didn't use half an hour?  Could you still win?

Harry and Gabe:
I pressed 1 hour when it was 11

Luke and Eve:
We think that you have to press 5 hours and 2 half hours for player 1 to win the game and 4 hours and 3 half hours for player 2 to win the game.

Dylan and Jacob:
We wrote that we did 1 hours and half an hour.

Thank you very much for these contributions.  The difficulty is to find a way of winning that ALWAYS works.

Rhys from St. Michael's on the Mount in Bristol, England, sent in this email:

With this strategy if you go second you are guaranteed to win.
The aim is to get to 7.30, then 9.00, then 10.30 - after that you have won.
If player one goes to 6.30 you would go 1 hour to 7.30.
If player 1 goes to 7.00 on their opening move you would then go half an hour to 7.30.

That is a fantastic strategy, Rhys.  I wonder if anyone can explain why it works?