Saturday, September 6, 2008

Theorem of Pants

I think pretty much everyone who reads my blog would also have read Anthony's Theorem of Pants that he posted on Facebook. But just in case you haven't seen it, it's the most beautiful theorem I have seen in a while. It goes something like this:

During our open house-less time up on FOSO (the floor in my dorm that i live on) us math/physics/engineering folks got a little restless. I started the original theory of pants, and with the help of other restless mathematically inclined people, finished it, making it actually pretty good. I now present you with "Anthony's Theory of Pants" (with special accompaniment by peanut gallery quotes and Special Pants Relativity)!!

This is the board the theory was developed on...its in our hallway...
*It must be noted that while this board is pretty to look at, there are some errors on it. These errors are fixed in this note.*


The amount of pants being worn is inversely proportional to the amount of pants that SHOULD be worn.

(Pants proper) = x / [(pants worn) +2]

where (Pants proper) is defined as the proper number of pants that should be worn
(pants worn) is the number of pants worn
and x is a number that reflects the appropriate level of pant-lessness.

Now we must further define x, as it is clearly ambiguous at the moment. The thing to notice is that x is a variable, not a constant. x varies according to the situation. For example: during open house, it is a very large number, meaning pants MUST be worn.

x(g) = (g + q) / q

where g is the number of girls allowed on the floor
and q is the quantized charge on an elementary particle.

Therefore, if we evaluate x only using units (in order to discover the units of (pants proper) we get

x(g) = (girls + coulombs) / coulomb
x(g) = girls/coulomb + coulombs/coulomb
x(g) = girls/coulomb

if we then plug x(g) back into our original equation, we get

(Pants proper) = x / [(pants worn) +2]
(Pants proper) = [girls / coulomb] / [pants]
(Pants proper) = [girls / coulomb] * [1 / pants]
(Pants proper) = girls / (coulomb * pants)

We can now use this formula to properly define the proper number of pants to be worn at any time on our floor.

pants values:
these values can be added and manipulated any way you want (example: a proper pants rating of 1 means it is possible to just wear four pairs of boxers...its just not recommended...)

What's beautiful about this theorem (to me) is the pantslessness function x(g). If there are no girls allowed on the floor (g=0), then x(0)=1. Thus, (pants proper)=1/(pants worn+2), meaning that if you are wearing any pants at all, pants should be removed to satisfy the relationship.

However, if even one girl is allowed on the floor, x(g) instantly becomes immensely large (as g is constrained to the set of non-negative integers). Thus, when girls are allowed on the floor, many pants must be worn by all pants-wearing-capable members. It's so beautiful. Nice job Anthony.