Stop the Clock

We describe the winning algorithm for the game where moves of only 1/2 hour or 1 hour are allowed.

A number of you pointed out that a key time was 10.30 - if we land on 10.30 then our opponent must either move to 11.00 or 11.30, and either way we've won. So landing on 10.30 means we're guaranteed to land on 12.00.

The idea is to backtrack, and list a number of key steps around the clock. Landing on 9.00 means we're guaranteed to land on 10.30 if we want to, using the same argument above. Similary landing on 7.30 guaranteeslanding on 9.00, landing on 6.00 guarantees landing on 7.30 etc... the full sequence of key times is

1.30, 3.00, 4.30, 6.00, 7.30, 9.00, 10.30

Any player who lands on one of these times will win for sure, since they will always be able to move onto the next key time, whatever the other player does, until they get to 10.30.

It follows that if player 1 starts at 3.00, 6.00 or 9.00 then player 2 can force him to lose, but if he starts on any other time, then on his first turn he can move onto a key point and make sure he wins.

See if you can now extend this idea to the full game, where players can move at 15 minute intervals.