There's some information missing - does he have to go to an adjacent corner, or for example, can he go from corner 2 to corner 6? Also, I don't see what you're trying to optimize - do you want him to divide his time as equally as possible between the corners, or do you want him to be as unpredictable as possible?

If you want him to be unpredictable, you have to choose randomly with equal probabilities from among all the possibilities. I think that can be demonstrated by example. Suppose you require him to move at each hour - he can't stay in the same place. Then someone (a criminal?) could be completely safe by always moving to the place he moved from. No matter what scheme the policeman cooks up, the criminal can better his odds because he can (at least partially) predict where the policeman will be.

The problem only becomes interesting if he has to move to an adjacent corner, and you want him to spend as close to 1/6 of his time as possible on each corner.

Here is one way to spend exactly 1/6 of his time on each corner. Starting from corner 4, choose from the following options with equal probability:

1. Go to corner 3, stay there 5 hours, and go back to corner 4.

2. Go to corner 6, stay there 5 hours, and go back to corner 4.

3. Go to corner 1 (1 hour), corner 2 (3 hours), corner 5 (1 hour), then back to corner 4.

4. Go to corner 1 (2 hours), corner 2 (1 hour), corner 5 (2 hours), then back to corner 4.

5. Go to corner 5 (2 hours), corner 2 (1 hour), corner 1 (2 hours), then back to corner 4.

If you take all five tours, you spend 5 hours in each corner.

I'm not sure that answers your question. Please post again if you still need help.