Finding equilibrium with two dependent variables I was thinking while in the shower. What if I wanted to set two hands of the clock so that the long hand is at a golden angle to the short hand. I thought, set the short hand (hour hand) at 12, and then set the long hand (minute hand) at what ever is a golden angle to 12. I realized tho, that the short hand moves at 1/12 the speed of the long hand, so I would need to move the long hand 1/12 whatever the long's angle from 12 is. However, doing this will effect the angle between the two hands, so I would need to more the long hand further. But again, I would need to move the short hand further because the long hand was moved further, and this creates a vicious circle until some kind of equilibrium is found.
Rather than doing this through trial and error, there must be a way to solve this problem mathematically. Maybe with an equation? My math isn't very good, but I'm very interested in math, so this is why I'm posting this question.
Could someone give me the solution to this problem while explaining it very throughly?
Thanks for reading. I'm interested in seeing what answers pop up on this question. :)
PS: Please help me with the tags for this question, as I don't know which tags would be appropriate.

Start at 0:00 where the two hands are on the 12. You now that the long hand advances 12 times faster than the short one so if the long hand is at an angle $x$, the angle between the two hands is $x - x/12 = 11x/12$. So you just need to place the long hand at an angle $12 \alpha/11$ so that the angle between the two is $\alpha$.
