Bifurcate the world into the 3x3=9 chances of which door the prize is behind and which door the contestant guessed and write down the permutations. These are two independent random events and the permutations are equally likely with a 1/9th chance each.
Now for the cases where the contestant guessed the door correctly -- "11", "22", and "33" the right move is to stay and it doesn't matter which door Monty Hall reveals.
For the cases where the contestant didn't guess the door correctly -- "12", "21", "13", "31", "23", "32" the right move is always to switch (Monty Hall will have always revealed the third door in each permutation).
The probabilities of each are 1/9th so they add up to 1/3 chance of failing if you switch and 2/3 chance of succeeding if you switch. So always switch.
If you want to complicate things you can add the door that Monty Hall shows as a third option, but you need to be careful about the probabilities:
"112", "113" - these have to add to 1/9th so each has 1/18th chance (50-50 chance MH picked one door or the other) and both fail if you stay, similarly for "221", "223", "331", "332", 6*1/18 = 1/3rd.
"123" -- MH is forced to pick "3" so this is just 1/9th chance of success, same for "213", "132", "312", "231" and "321". So this is 6*1/9 = 2/3rd.
One error would be writing down all the permutations and assuming they were equally likely, but "112" and "113" must sum to the odds of "11" which is 1/9th, while "123" is the same odds as "12" which is 1/9th.
Now for the cases where the contestant guessed the door correctly -- "11", "22", and "33" the right move is to stay and it doesn't matter which door Monty Hall reveals.
For the cases where the contestant didn't guess the door correctly -- "12", "21", "13", "31", "23", "32" the right move is always to switch (Monty Hall will have always revealed the third door in each permutation).
The probabilities of each are 1/9th so they add up to 1/3 chance of failing if you switch and 2/3 chance of succeeding if you switch. So always switch.
If you want to complicate things you can add the door that Monty Hall shows as a third option, but you need to be careful about the probabilities:
"112", "113" - these have to add to 1/9th so each has 1/18th chance (50-50 chance MH picked one door or the other) and both fail if you stay, similarly for "221", "223", "331", "332", 6*1/18 = 1/3rd.
"123" -- MH is forced to pick "3" so this is just 1/9th chance of success, same for "213", "132", "312", "231" and "321". So this is 6*1/9 = 2/3rd.
One error would be writing down all the permutations and assuming they were equally likely, but "112" and "113" must sum to the odds of "11" which is 1/9th, while "123" is the same odds as "12" which is 1/9th.