# Toilet Paper Paradox

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The Toilet Paper Paradox is a mathematical paradox that relies on the surprising properties of infinity and toilet paper. It was discovered in 1842 by mathematician Dr. Phil McGraw.

The paradox goes like this:

There are two people sharing a bathroom (more advanced variations feature several people, but to keep it simple we will use two). As they go through the roll of toilet paper, the amount of remaining toilet paper is a linear function as long as the amount of toilet paper remaining is greater than the toilet paper needed.

However, when the toilet paper roll gets down to the last few squares, the user will only use half of the remaining toilet paper as an excuse to not get a new roll. Despite the fact that the amount of toilet paper being used is not enough for anyone to actually be satisfied using, each person will gladly make do as an excuse not to go all the way to wherever the toilet paper is stored, and then go all the way back, due to Theodore Pooper's (TP's) Theorem:

Let f(x) be a function defining the length of route taken to get the toilet paper, and x be the length of the roll of toilet paper roll, where amount of toilet paper in need is greater than toilet paper present:

${\displaystyle \lim _{x\to 0^{+}}f(x)=\infty \;}$

Another more practical theorem states ${\displaystyle f(x)}$ is a function of how much toilet paper is left, where ${\displaystyle x}$ is how many times the toilet has been used. ${\displaystyle f(x)}$ is always less than ${\displaystyle f(x+1)}$, but function ${\displaystyle f}$ will never equal zero. In theory, the pair could go on forever pulling the toilet paper off strand by strand, but in practice, someone will eventually get off their ass (literally) and change the toilet paper.