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.

1,135 replies on “Urinal protocol vulnerability”

  1. I hate when there is an odd number of urinals and you end up next to the creepy guy that keeps looking over. As much as you try to ignore him he keeps looking away and looking back. Really dude?


  2. I like to walk up and pee at the urinal next to the guy who already looks awkward and make the awkward factor even greater


  3. I like to take the urinal directly next to someone and increase the awkwardness even more by making a comment such as…”hey thats a nice watch!”


  4. “hey, that’s a nice watch” does the trick, as does “wow, did you hit your thumb with a hammer?”

    I didn’t really examine the mathematical equations thoroughly, but the drawings indicate we are installing approximately twice the number of urinals required. By omitting every other urinal, the protocol is nullified.

    By the way, in Army training barracks the stools are side-by-side. Wish there was some government money available to study and establish a protocol for them as well. Not for me, I doubt I’ll ever use them again.


  5. You missed the most efficient configurations, which are a single urinal, (plus any number of closed stalls, of course). The next question is whether dividers between urinals can decrease the awkwardness factor, & if so, how big they need to be in order for most users to be willing to stand next to another user.


  6. i’m flabbergasted…i thought it was tough being female, but after reading this i now realise that mens lives are fraught with difficulties.

    as far as correlating periods to holidays – it isn’t worth the effort! it’s gonna happen no matter what the numbers say!

    playground clapping rhythms – that’s genetic codes that guys will NEVER understand! 😉

    and if guys knew what happened at slumber parties, we’d have to kill you!!


  7. Oh heck, I’ve been so busy with Mr-Pickett’s class I’ve forgotten to do my Maths homework for Mr Grimmet. I better do it or he won’t let me daisy chain on my sacking sampler tomorrow. I like Mr Grimmet, I sit on his knee while he threads my needle


  8. Honestly I don’t pay much attention to who is in the washroom when I enter. Ijust pick the nearest urinal and get out as quickly as possible, after washing my hands of course.


  9. Just remember best situation of 3;5;9 and worst of 4;7;13, since a toilet which afford over 20 guys is out of realistic


  10. Sie amk. I was so sorry to hear what you were going through since November….and seeing your Monday April 4 comic – I wonder if things are better or how you’re coping up with the illness in your family. Your comic strip has made so many, many of us laugh through the years and have a sense of humor about life through our tough times, wishing you strength during yours. vay ak.


  11. You are the most pathetic bunch of wimps imaginable. I always head for the closest urinal that is available, so that I’m always next to at least one guy unless the room is otherwise empty. I usually have a guy on each side of me. The only exception to this rule is that I avoid stinking urinals full of piss. Some guys are really pigs. In other countries, guys stand beside one another and piss against a wall. That’s how it was when I was growing up. In Australia, men stand next to each other on a metal grate and piss against a stainless steel wall. You can watch everyone’s piss mingling as the stream flows to a drain on the side. It’s customary to each a brief greeting such as “Hi mate (pronounced more like mite)” or “Were you at the footy last night?” Men are still men in Australia.


  12. Has the sequence of the best (the peaks) and the sequence of the worsts (the valleys) been identified previously or are you the first one to describe them? Should be interesting to see where else in nature (and in the subset humanity) those sequences arises.


  13. People should just build small partition walls between the stalls (no need for cubical like the ones in female toilets, just small little pieces of wood to prevent the awkwardness. No one is going to see anyone’s anything.) and then everyone can pee in peace.


  14. Looking at the examples there is something flawed with this logic. We men can always make urinals over 50% efficient if person 3 takes a sacrifice for the team.

    Example 2 person 3 goes straight in the middle so only 3/7 urinals are used if he moved over just 1 urinal closer to either person 1 or 2 then a number 4 could have been able to get in and still have the 1 urinal separation.

    At first it may seem weird like wtf why is this guy coming closer to me but he is still flowing the main rule 1 urinal distance

    Its all about that third person if they go in the wrong spot they messing it up for everyone


  15. out of say 5 urinals, guys will go to 1 5 3 first, then go to 2 and 4 depending on the space between the urinals. Some bathrooms have a foot or more of space between urinals and others are flush against each other. so we can get 100% efficiency of a urinal row if it isn’t set up to have people touch shoulders at max capacity.


  16. I prefer my method, stand as far back from the Urinals as possible and try the distance shot. Awkward for those around you and adds some serious presure to aim right 😉


  17. This is bloody hilarious. I also think the alcohol factor should be considered into the equation. More alcohol consumed over the evening reduces the ‘aim’ of the stream. As a result, more than one urinal space in between each participant should be a constant.
    Also, from my experience, most guys tend to use the actual toilets if only one person is at the urinals, irrespective of how many there are.
    How about this, use the closest urinal/trough, do your piss without cutting it short and spilling some on your pants because you dont want your wang out in public for too long.
    We are men! This evolutionary ability has set us well above women since the dawn of time. Embrace it!


  18. Nice Post, but The best way to avoid awkwardness in designing urinals is put stone slab of suffcient height between the urinals. Max efficency, minimum awkwardness


  19. I never thought someone would come out with some mathematical analysis for this! Good job!

    I would definitely love to read some female related issues, like their interaction with men, or some behavior change in a person, when a special someone is present and when not!


  20. This is bloody brilliant. With mathematical certainty, we can determine the awkwardness quotient.

    Love XKCD.


  21. 7 really isn’t that bad… allow me to demonstrate
    * * * * * * *
    1 4 3 2.
    Now is that really the end of the world?


  22. In the case of 7 urinals where I am the 3rd person, I use a modified Protocol designed to minimize the chance of awkwardness. If the 3rd person takes urinal 4, then there is a 50% chance that the 4th person will choose a spot immediately adjacent. If instead, the 3rd person chooses urinal 3, then assuming the 4th person is a rational actor, he will choose 5, and there will be 0% chance of awkwardness.


  23. Is it bad I go to the immediately adjacent urinal to the most recent person to begin peeing and….wait for it….

    Pee in their urinal.


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