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December 16, 2002

The algorithm of toilet paper

A few days ago I wrote about Danny O’Brien’s algorithm for washing dishes — illustrating the weird complexity of everday tasks.

Andrew Wu posted a comment pointing this totally excellent site by Don Norman — where Norman studied the algorithm of how we use toilet paper!

Norman had the usual problem we all faced: We never replace the toilet paper until it’s dwindled to the end of the roll. And if that happens to be your final roll — you’re out of luck. So he tried to solve this problem by installing two toilet-paper holders side by side, the same way public bathrooms have them. Interestingly, it didn’t work:

We discovered that although we now had two rolls instead of one, the problem was not solved. Both rolls ran out at the same time. Sure, it took twice as long before the rolls emptied, but we were still stuck with the same problem: no more paper. We had discovered that the switch to two rolls meant we had to use more sophisticated behavior: the algorithm for tearing of paper mattered.

After some self-observation and discussion, we discovered that three different algorithms were in use: large, small, and random.

Algorithm Large:Always take paper from the largest roll.

Algorithm Small:Always take paper from the smallest roll.

Algorithm Random:Don’t think — select the roll randomlyWe had assumed that Algorithm Random was most natural. After all, we had bought the dual-roll holder specifically so that we wouldn’t have to think. But were our selections truly random, we would chose each roll roughly equally, so they would both empty at the same time — or close. Algorithm random is not the one to use. To use toilet paper requires thought.

Our self-observations revealed that we really didn’t use the random algorithm — people are seldom random. The most natural: that is, we soon discovered, was to reach for the larger roll. Alas, consider the impact. Suppose we start with two rolls, A and B, where A is larger than B. With algorithm large, paper is taken from A, the larger of the two rolls until its size becomes noticeably smaller than the other roll, B. Then, paper is taken from B until it gets smaller than A, at which point A is preferred. In other words, the two rolls diminish at roughly the same rate, which means that when A runs out of paper, B will follow soon thereafter, stranding the user with two empty rolls.

Algorithm small turns out to be the proper choice. With algorithm small, paper is always taken from A, so it gets smaller and smaller until it runs out. Then paper is taken from roll B, which is full size at the time of the switch.

Yikes. We never realized that you had to be a computer scientist to use toilet paper.

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I'm Clive Thompson, the author of *Smarter Than You Think: How Technology is Changing Our Minds for the Better* (Penguin Press). You can order the book now at Amazon, Barnes and Noble, Powells, Indiebound, or through your local bookstore! I'm also a contributing writer for the *New York Times Magazine* and a columnist for *Wired* magazine. Email is here or ping me via the antiquated form of AOL IM (pomeranian99).

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