I've cracked it. Hopefully I will eventually be able to generalize the technique.
The Monty Hall problem is not a problem in the direct sense. To work out the probabilities mathematically is pretty trivial. You can even draw all the chains of possible events and just count the outcomes.
The problem is that it starkly highlights a serious logical deficiency in most human brains.
The general solution used is thus;
We have three doors. We know one of them is a goat. Therefore, there's a 50% chance that one of the closed doors is a car.
As far as it goes, this is true. However, since it gets the wrong answer, it must be throwing out some relevant information.
Instead, examine the case of choosing to switch. If the above analysis were true, it should net the same probability as choosing to stay.
This is what happens;
Choosing a door eliminates it. Monty Hall then eliminates a goat. The chance the first elimination eliminated a car is 1/3. The chance Monty eliminates a car is 0. Therefore, the remaining door has a 2/3 chance of being a car.
Now I've worked out what I should have done in the first place, I can work backwards to see how I should have thought to do that in the first place, and thus plug this troubling logical hole.
August 10, 2008 update :
Here's a link to something I watched. I successfully re-analyzed the problem successfully, and so I declare this self-patch a success.