The minecart goes the wrong way when dispensed. This can be fixed by dispensing the minecart onto a flat track with a block against the other end, or by putting the dispenser at the top of the track.

That would solve the puzzle automatically once the player enters the dungeon.
No, the real way to fix this, would be by adding a second Mine Cart to the Dispenser.
See #33

@Robijnvogel Couldn't we replace the minecart with a redstone-only circuit that does the same thing?

Of course we could, but that wouldn't be half as interesting.

I didnt think this was a puzzle, as the only input I could identify was the pressure plate. Are those buttons connected to anything? they didnt seem to be.

Since the buttons are there, wouldn't the pressure plate be the last thing you'd think of pushing? :)

Ah, I see. So you are supposed to press it twice. While clever, I could tell right away that the buttons were superfluous due to their placement not allowing redstone connectivity. I would suggest a subtle adjustment to make them more plausible.