Urinal protocol vulnerability

When a guy goes into the bathroom, which urinal does he pick?  Most guys are familiar with the International Choice of Urinal Protocol.  It’s discussed at length elsewhere, but the basic premise is that the first guy picks an end urinal, and every subsequent guy chooses the urinal which puts him furthest from anyone else peeing.  At least one buffer urinal is required between any two guys or Awkwardness ensues.

Let’s take a look at the efficiency of this protocol at slotting everyone into acceptable urinals.  For some numbers of urinals, this protocol leads to efficient placement.  If there are five urinals, they fill up like this:

The first two guys take the end and the third guy takes the middle one.  At this point, the urinals are jammed — no further guys can pee without Awkwardness.  But it’s pretty efficient; over 50% of the urinals are used.

On the other hand, if there are seven urinals, they don’t fill up so efficiently:

There should be room for four guys to pee without Awkwardness, but because the third guy followed the protocol and chose the middle urinal, there are no options left for the fourth guy (he presumably pees in a stall or the sink).

For eight urinals, the protocol works better:

So a row of eight urinals has a better packing efficiency than a row of seven, and a row of five is better than either.

This leads us to a question: what is the general formula for the number of guys who will fill in N urinals if they all come in one at a time and follow the urinal protocol? One could write a simple recursive program to solve it, placing one guy at a time, but there’s also a closed-form expression.  If f(n) is the number of guys who can use n urinals, f(n) for n>2 is given by:

The protocol is vulnerable to producing inefficient results for some urinal counts.  Some numbers of urinals encourage efficient packing, and others encourage sparse packing.  If you graph the packing efficiency (f(n)/n), you get this:

This means that some large numbers of urinals will pack efficiently (50%) and some inefficiently (33%).  The ‘best’ number of urinals, corresponding to the peaks of the graph, are of the form:

The worst, on the other hand, are given by:

So, if you want people to pack efficiently into your urinals, there should be 3, 5, 9, 17, or 33 of them, and if you want to take advantage of the protocol to maximize awkwardness, there should be 4, 7, 13, or 25 of them.

These calculations suggest a few other hacks.  Guys: if you enter a bathroom with an awkward number of vacant urinals in a row, rather than taking one of the end ones, you can take one a third of the way down the line.  This will break the awkward row into two optimal rows, turning a worst-case scenario into a best-case one. On the other hand, say you want to create awkwardness.  If the bathroom has an unawkward number of urinals, you can pick one a third of the way in, transforming an optimal row into two awkward rows.

And, of course, if you want to make things really awkward, I suggest printing out this article and trying to explain it to the guy peeing next to you.

Discussion question: This is obviously a male-specific issue.  Can you think of any female-specific experiences that could benefit from some mathematical analysis, experiences which — being a dude — I might be unfamiliar with?  Alignments of periods with sequences of holidays? The patterns to those playground clapping rhymes? Whatever it is that goes on at slumber parties? Post your suggestions in the comments!

Edit: The protocol may not be international, but I’m calling it that anyway for acronym reasons.

1,068 thoughts on “Urinal protocol vulnerability

  1. f(n) = 1 + 2^(int(lg(n-2)-1)) + max(0,(n-(3/2)(2^(int(lg(n-2)))-1)
    outputs 4.5 for n=5

    Am I just not interpreting it correctly?

  2. Oops, I was using log base 10, not log base 2. It checks out with log base 2. :)

    So the important take home message is that if you are building a men’s restroom and trying to decide between 5, 6, or 7 urinals, you should go with 5 because only three will be used at a time anyway.
    And if you are looking anywhere in the range 9-13 you should just go with 9. I feel like someone should really make this public knowledge…

  3. How about having partitions in the wall about 7 ft pprox…between every urinal, in that way I guess in that sense awkwardness is bearable?!

  4. I remember a situation… there were 4 urinals. I come in, there’s already a guy. So I take the last possible urinal. Then another guy comes in and goes directly between us. Nobody spoke.

  5. What about those little walls that separate each urinal? You can build as many urinals as you want.

  6. @not_essence — the other key protocol of urinal etiquette is that nobody at a urinal should talk. Ever.

  7. I don’t know why trough / waterfall style urinals are not more common in western countries. We have lots of these in Hong Kong public toilets, so when there are 3 men peeing and a 4th comes, he just goes between two of them and the middle guy moves along.

    Also, sometimes I happen to need to pee at the same time as another male family member. We go to the toilet separately but if there is an awkward situation already, then it is OK to use two urinals next to each other. I wonder what this does to the math.

  8. Really? When I am using the same bank of urinals as a family member, I choose the farthest one or opt for a toilet stall. You guys must be close.

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  11. It seems to me that it would be more cost-effective to install 3 urinals with plenty of space between each, rather than installing 5 in the same space.

  12. In my experience the protocol breaks down at sporting venues during peak times. Only really once all of the stalls are taken though.

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  14. Even with 8 unrinals, #3 could have taken spot 3 and left the other open, then #4 comes in and Checkmate Bitches.

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  16. This is all nonsense. If there’s an empty urinal, use it. If you’re shy, hold it in or stay home.

  17. Perhaps it’s a mathematical writing convention to which I’m unaccustomed, or my eyesight is worse than I thought, but the formula appears to me to be missing something within the “MAX” function. Probably a pair of parentheses? (that “int lg” part of the formula appears to be missing one) Perhaps a comma somewhere? (that doesn’t seem likely) Should the “-1″ be part of the exponent (it doesn’t appear to be written that way). It also looks odd to see a formula written as “(3/2) * 2^exponent” when one could simplify that to “3 * 2^(exponent – 1)” Am I overlooking something?

  18. what does xkcd mean? your blog have no about page so I dont know what does it mean. You have a nice blog… and i really enjoy to read it.. Keep posting!

    Thank you!

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  20. There’s also the Door Proximity Effect that must be taken into account in this formula: If the door isn’t directly behind the bank of urinals, and is instead to the left or the right, it must be treated with slightly less priority than an occupied urinal.

    For example, in a bank of five urinals, The first person would take urinal 5, the furthest from the door. The second person, in the case DPE, would not choose urinal 1, as described under ideal conditions, but instead urinal 2, providing himself with an adequate buffer from the door as well as the first person. His choice of urinal 2 instead of urinal 3 stems from the slightly higher Awkwardness rating of the person than the door.

  21. This also applies to automatic wifi channel selection in most routers.
    I recently had wifi connection problems and discovered that the channel occupancy was following the same pattern. There are about 11-13 wifi channels, the first couple of routers takes #1, the second bunch takes #11, the 3rd takes #6… It appeared that everybody in my vicinity used the default wifi setting that includes automatic channel selection. So, I just forced my router to take #3 (totally free) and was fine until many many wifi devices would ever be around (I could even lower the emission power of my router).

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  23. A Stack Overflow question prompted me to revisit this post and explore how to create optimal groups out of a suboptimal number of urinals, and I believe a more general case solution than “take one a third of the way down the line” is “take the Nth urinal down the line where N is the largest optimal number smaller than the number of urinals. In the worst case, I believe this is equivalent to your rule, but does better in average cases, such as 20.


  24. I have a 100% Packing solution.
    Get rid of urinals, only have stalls.

    If we assume that the average stall user is not horrendously overweight, stall spacing could easily approach the same the same spacing that a single urinal takes up.

    In fact, if you deny men the ability to stand up and urinate, you could easily pack even more “seats” in with the added benefit of not having the bathroom smell like someone pissed all over the floor.

    The idea being putting in a central column of stalls that are only as big as the toilet. Removing the obsolete “wooden door” with a folding door (or, for better security) a “folding steel door” that rolls from the side of the stall to the front) would make entrance and exit easy.

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  26. So, you guys do all this and still manage to go to the restroom faster than women at crowded events? I’m guessing that when long lines would result otherwise, y’all just let awkwardness occur.

  27. Wanna know how to place urinals in such a way that it’s impossible to create awkwardness unless some aspiring urinator really wants to try? This:
    Just put walls between each station!
    Another way, although I’ve only seen this in practice once, is to employ a sort of long urinal-trench, so that however many guys there are that need to go at the same time, they can all just stand at optimal distance from each other.

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  30. The above mentioned function is not defined for n = 2, since lg(0) is undefined.

    This is false. If there are two urinals, one guy could use it without violating the awkwardness rule.

  31. This algorithm fails to account for the orientation of the urinator and the hotness of the other urinators in progress.

  32. It would be interesting if you took into account walls separating 2 urinals. Another interesting aspect is that of “psychological awkwardness”, for example, 5 urinals provides an optimal configuration, but say we removed the middle urinal. Does awkwardness still ensue between the 2nd and 4th urinals?

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  38. There needs to be an ISO protocol created for this. I think woman design the bathrooms, every guy is familiar with this issue. What I want to know is why they do not introduce dividers between the urinals to create some personal space; or why some bathrooms have two urinals.

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