The Five Schools of Mancy
The final of the Five Schools of Mancy in the Infinity Hotel: The Hypermath Cult. Known to their rivals as the Logarithmancers.
What exactly is Hypermath? It definitely involves drugs. The above picture can give you an idea.
Basically, it's all about making weird psychedelic drugs and using them in very particular ways to access parts of the Hotel nobody else can, although the Deep Hotel still alludes their proofs.
(For those of you new to the series, The Infinity Hotel is a place with a countably infinite number of rooms, all of them filled, yet with paradoxically room for more. More than that, it's an epic-level adventure location. There's a Mystic Elevator that can take you to any room number that you want, provided you can express the entire number. People with people-brains just cannot express some numbers, but other things can.)
The Deep Hotel encompasses the parts inaccessible to human brains, like the number double beyond what one gets if they spent 4000 years programming computers to program computers to generate Eight over and over (88888888888888888888888... etc.), which doesn't even infinitesimally get close to the room numbers of the Deep Hotel. These numbers still elude the Hypermath Cult.
What they've found out in the meantime, however, is perhaps more significant: it's that there's more than one Hotel.
There's an Imaginary Hotel, a Complex Hotel, a Fraction Hotel and a Negative Hotel.
Most people visiting the Infinity Hotel never come to hear about this places, and it's probably for their benefit. The non-Natural Hotels are weird at best, and you can't confine yourself to Natural Reality if you want to experience them. You gotta use drugs. MATH Drugs.
Table doubles as "What is this Hypermath Cultist carrying?".
MATH Drugs of the Hypermath Cult
1. Everwake - Tastes like fire. Don't need to sleep for a day. Accumulate 'sleep debt', and following day must sleep double the amount of time. Each time sleep put off the 'sleep debt' doubles.
2. Beerception - Pleasantly hoppy. Chugging causes perception of time to slow down dramatically. Experience a year in a single minute, or 10 times that if you chug two.
3. Warp - Causes nose to bleed. Teleports user to random viable space on current Plane.
4. Euler's Pie - Blueberry flavor by default, many options for taste available. Eater knows the exact surface area, volume, and dimensions of everything in immediate area.
5. Frac - Leaves blue track marks. One may infinitely telescope/magnify their vision.
6. Break - "I'm taking a break." Common saying. Inhaling halves the width of user's body. After a stick or so one will be nice and flat, perfect for the Fraction Hotel. Also good for being left alone.
7. PolyNom - Gives splitting headache. Incapacitating gas, used for combat/imprisonment. Those inhaling must solve polynomial equation before doing anything else, including breathing.
8. Mirror Mirror - Good for skin, side effect of deafness. When used in front of a mirror, allows access to the Negative Hotel, the Anti-World with its anti-versions of everything. The Anti-Philosophical Razors can be found if used in Halfway.
9. iStoned - Turns eyes into psychedelic color portals. Allows one to traverse towards the perpendicular Imaginary Hotel, with rainbows and unicorns! (Note: Unicorns may not be friendly.)
10. Konk - Unicorn distillate, smoked in water pipe. Allows one to traverse into parallel Complex Hotel. Not suitable for humans. Disillusionment of concept of Self likely. Won't be able to stand Natural reality anymore. Will need more Konk.
True to its name, The Hypermath Cult is a cult. There are ten Levels to the Cult, gained by making contributions of money, drugs, and mathematical proofs. Each level bestows upon its wielder Powers of Math.
At level 2 the Cultist will gain powers of Logarithmancy - the ability to shred HP and incoming damage through a logarithm equal to their Cult Level. They'll start doing simple mathematical proofs and gateway drugs (1-3 on table above).
At level 4 the Cultist will be shown the wonders of the Imaginary Hotel, and gain the ability to summon things from its plane. These Cultists tend to be the most dangerous, as they have little ability to control their newfound power of summoning anything they imagine.
At level 6 the Cultist will come to know the Negative Hotel, and be taught its dangers, chief of which is DON'T TOUCH YOUR DOUBLE. YOU WILL EXPLODE.
At level 8, which very few obtain, the Cultist can be shown the Complex Hotel, and they will be permanently changed. These Cultists start wearing orange robes, tall pointy hats, and masks of gold, to hide their strange features from the Natural World.
There is only one level 10 Cultist, and that is Timothy Amherst, the Esteemed High Mathematician of the Hypermath Cult. He has the ability to humble gods and survive point-blank nuclear blasts with his Logarithmancy. A very tall man robed in blue, with a pointy blue hat and a serene bronze face mask that's said to protect others from his mystifying gaze.