Toilet Paper Folding Algorithms

If I recall correctly, I read somewhere that the average American uses 7 sheets of toilet paper per toilet visit. I tend to lean on the cleanliness-obsessive side, so I usually use 16-20 sheets per visit. Yeah, I know, in this aspect I’m more wasteful than the average American, which would be pretty shameful, except that I’m sure that the many of you who have seen the alarming variety of material I have used as scratch paper can hopefully give me credit for making up for toilet paper overuse in scratch paper conservation.

Anyway, most rolls of toilet paper I have seen have pre-divided segments, partially perforated, such that each piece is roughly square and one can fold and tear the paper easily and neatly. This is the type I have seen in most homes, as well as in New House. In public facilities, however, one can frequently find the lesser-quality toilet paper (frequently of the Kimberly-Clark brand) that is noticeably narrower and also doesn’t have pre-perforated pre-designated sheets. MacGregor happens to also have Kimberly-Clark paper, so I’ve started to have to use such paper regularly. This type of toilet paper has given me qualms, since it’s significantly harder to be neat with this paper than with the pre-perforated kind, especially with end-rips. What’s more, when I fold the paper, I have to guess how much to fold it by, which often ends up producing an awkward stub of paper not long enough for another fold around that I’ll have to tuck in next to the previous fold.

I eventually found a solution to the issue of non-integral length in sheets by just ripping out a large portion, folding it in half, then in half again, and again, until it is the closest rectangle it could be to a square. I started using this approach to folding such free-form toilet paper. After a while, though, I realized that it is also a good idea to fold pre-perforated toilet paper in this fashion. Specifically, a less total number of folds is needed in the pre-set paper length case, because most folds fold at multiple locations along the strip of paper. Algorithmically, sequential toilet paper folding occurs in O(n) folds, whereas binary toilet paper folding occurs in O(lg n) folds. Dealing with a more imperfect type of toilet paper caused me to realize a universal improvement to toilet paper folding efficiency. This is just another example of the importance of perspective-enhancing alternatives.

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