Having finally managed to get Graham's medicine all ready for his annual checkup, the pair of you set off to the edge of the Everfree Forest to look for Graham. Eventually, after a brisk walk involving small talk and fending off several hungry manticores, you arrive at your destination. And, in a moment that would make you want to face-hoof at the sheer improbability of it all, you find that conditions are perfect to hit you with yet another puzzle.
Graham, as Fluttershy explains to you, has taken residence in a series of caves, inside a mountain. However, by a frankly startling coincidence, the mountain appears to be shaped as a layered pyramid of sorts. Nopony knows exactly how the mountain came to be that way, but current thinking favours extremely bored pegasi who wanted to see if they could just use rain, wind, and the march of time to create something so neat it couldn't have happened by chance. Or aliens. probably aliens.
Anyway, this pyramid of caves is arranged so that the inside of the cave on the top layer is one room, the layer underneath containing two rooms, the third containing three rooms, and so on and so forth. Graham's place of residence on the mountain is on the floor that satisfies the following conditions: all the rooms in the caves have exactly three doors, that lead to another room, all except the one where Graham is holed up in, which connects to only one. Fluttershy assures you that you won't need to climb any slopes, ledges or stairs to get to where Graham is, as being a pegasi, she can fly you up herself (probably), and will need to be present for his checkup anyway. Also, there are no passages connecting rooms.
You're going to have to do it as discreetly as possible though. Graham has more sensitive hearing than you'd think, and if you stomp around the caves, he'll surely try to escape his doctor's appointment and become a bit violent. Therefore, you'll only have time to search one of the layers of the pyramid cave before he detects your presence. This does absolutely nothing to reassure you that Graham won't be something harmless like a bat or something, but you feel that a bit of logic should compensate nicely.
You ask why Fluttershy doesn't just visit Graham on her own, but she explains that usually Angel Bunny goes with her, but this year he said that he had enough years taken off his rabbit life accompanying her to these visits, and no amount of cherries on top his lunch will ever make up for it.
Well, darn. Fluttershy, unfortunately, didn't draw up a map for you to follow, but nonetheless, you resign yourself to the unpleasant sounding job. So where the heck is Graham (whatever he is)?