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. This would work well at my school where there are 4 bathrooms. Each with 2 stalls and 2 urinals for maximised awkwardness at any time when there is > 1 person present.

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  2. does this apply no matter how far apart the urinals are? could building planners overcome the problem of money wasting, un-used urinals by simply increasing urinal spacing and thus reducing awkwardness, allowing consecutive urinals to be used at the same time? (similar but cheaper than installing “half stall” style partitions) as a female i don’t know if this would be acceptable nor what the optimum spacing would be if it were, it just seemed like a logical solution…

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  3. >What a fucking waste of time.
    What? posting your comments?

    The solution – is barriers b/w each urinal, so you don’t have to stare at the wall to keep the other dude out of your peripheral vision.

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  4. “Hacking” the protocol in the described manner, and always taking, say, the third stall of seven if you’re first, is not optimal. It ignores any estimates of the likelihood of a second, third, etc. guy coming in to pee. If the expected number of simultaneous guys who need to pee is low, taking the third of seven is suboptimal, as you’ll on average be closer to other guys than you need to be. Awkwardness is dependent on the full distance between guys, not merely whether there’s one or no urinals in between.

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  5. I think there may be anti-pattern worth mentioning too (I have tried to describe this to my wife several times – and even had to flag this to young nephews).

    I am not homophobic, but still, there is no denying certain creepy ‘George Micheal’ situations you are faced with when entering a urinal in say, a public park.

    There are times when you enter a urinal and clearly the Awkwardness protocol has broken down – in fact it seems the Awkwardness has dissipated completely – and as you passed nobody going out on your way in (in fact now you think about it, you didn’t see anyone coming in either …).

    The rule here is clear. If you cannot pee without Awkwardness (in the case of the midddle 3 of 5 being taken, or worse, the first and last 2 of 7 being taken – turn ’round and leave.

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  6. What? No. This isn’t international protocol at all. You don’t use one of the end ones unless you have a small dick. Other than that, sure you try to remain an equal distance from anyone, but since you’re not a homophobic arse^H^H^Hmerican you use one of the others (end in preference to between two others) if that’s your choice.

    Anyway, if you’re at a classy establishment you’re pissing up against a big stainless-steel wall rather than into a small target anyway. What’s with those girly washbasin-like things?

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  7. i wonder how urinals that are lower for kids affect this rule. often the first urinal is a short one. does the middle urinal then become the second choice instead of the first urinal, which is less mature because it’s a short one?

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  8. If they simply put up those little divider walls between urinals then you could have 100% efficiency and 0% awkwardness.

    Since all males know this and most construction is done by males why do we still have wall-less urinals? or worse yet those troughs where everyone just lines up and might as well be hand-holding will taking a pee.

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  9. The formula for f(n) can be simplified.
    If n-2 = 2^k + l with k,l natural numbers and l in {0,1,…,2^k-1}, then
    f(n) = max(2^(k-1) + 1, l + 2).

    This formula immediately explains all characteristics of the packing efficiency graph. And we can easily calculate that the maxima at n=2^k + 1 have packing efficiency 1/2 + 1/(2n). And the minima at n=2^k+2^(k-1)+1 have packing efficiency 1/3 + 2/(3n).

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  10. A possible female situation I have observed is the choices N women make when ordering from a menu. If the first woman to order chooses “a salad, but can you make it half size and hold the dressing?” as her main entre, then the women who follow directly are tacitly discouraged from ordering say, “The Buffalo Burger topped with Chili and a side of Fries.” There is a cooling off period after the first round of ordering and some reshuffling occurs, for example, if ordering appetizer and entre at different times, or desserts later. What is really going on?

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  11. Over here in Oz, there’s only one urinal. Might be a couple of metres wide, might be 10. You have infinite choice of position. Some choose spacing, some choose… closeness.

    Except in those weird places where they have separate thingos like in your diagram. I don’t know where they got them from. The US, or Germany, I guess.

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  12. Cmon, any serious urinal selection algorithm has to take in to account proximity to the door in the rare case the awkwardness requires you to flee.

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  13. You really should include the alcohol factor in your formula. The alcohol greatly affects the awkwardness factor and may help achieve 100% effectiveness. Or not being able to calculate may increase the awkwardness… anyway you got my point.

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  14. A similar protocol applies to women in stalls. You pick the one furthest away from anyone else.

    Also, if you’ve just taken a big dump, had a fartathon or rustled a tampon wrapper, most women are highly embarrassed. So most sit in the cubicle and wait until the other women leave before coming out. Thus, it’s protocol to finish your business quickly and not primp in front of the mirror if you’ve just heard someone take a big dump in the ladies.

    If you DON’T finish your business in time and the tampon-rustler flushes before you, it’s courtesy to wait till they’ve left before exiting to wash your hands.

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  15. I thoroughly believe that the protocol could be modified so that the third person always chooses a urinal that’s one urinal away from one of the two original ‘members’ (ahem). Then, the next person chooses a urinal two away from the third person, and so on until you either reach n/2n efficiency, or n/2n+1 efficiency, which, for an infinite number of urinals (truly a sight to behold I would think), approaches n/2n anyway. Anyway who occupies a space with two urinals between him and both members to the left and right is truly a ‘dick’ (ahem again) and deserves the piss beaten out of him.

    That efficiency with this modification:

    1 – 100%
    2 – 50%
    3 – 66%
    4 – 50%
    5 – 60%;
    6 – 50%
    7 – 57%
    8 – 50%
    9 – 55%

    And so on, approaching 50% in general.

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  16. This all goes out the window on a Washington State ferry when faced with a 4-8ft steel trough in heavy waves. Trying desperately not to cross the streams as everyone lurches side to side with the rocking of the boat.

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  17. This subject is so important how much the genetic study of the menstrual cycle of the blue butterflies that inhabit the setendrional region of the Island of Madagascar.

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  18. To prevent the miss-usage (ie: one person turning a good row into two bad) there should be lightsensors installed to the urinals who follow the same formula and checks if the person entering chooses the right urinal. If not, a spray of saved up pee will be emitted from the urinal upon the offender.

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  19. Urinal dividers pretty much solve all the Awkwardness problems, and inefficiency problems. That or they should just install half the number of urinals and space them 7 feet apart. Washroom designers are stoooopid. But why, why, oh WHY are there so many big mirrors across from the urinals? That creates Awkwardness too without privacy dividers.

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  20. What about the guy (and there is ALWAYS THIS GUY) that insists on standing next to you. Worse, he usually wants to talk to you as well. I pee on his shoes. Its about all you can do to shorten the awkwardness.

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  21. I am surprised that no one has mentioned the thing which I find most awkward in women’s washroom design. Specifically, that many new washrooms have spring-loaded stall doors that swing shut on their own. This makes it impossible to tell which stalls are occupied without either attempting to look under the door, or pushing lightly on the door, both of which are extremely awkward.

    Last weekend, I entered a new washroom in a mall and lined up behind two other women who were waiting for stalls. I waited about ten minutes before discovering that most of the stalls were empty; the women ahead of me hadn’t checked, and I had assumed that they had.

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  22. The protocol that works best is just to use alternating stalls that way the packing is guaranteed to be at least 50%. Though my own personal algorithm involves entering a queue for the stall.

    Perhaps the most effecient layout is just to alternate urinals with stalls, that way no one ever has to look at another person while ejecting waste.

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  23. you could minimise awkwardness for say n=13 by reconfiguring a line of urinals into a U shaped 5,3,5 and the guys in the corners would be at right-angles to each other. this would allow for 8 guys – 53.3%

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  24. A similar problem is seating in movie theaters. It seems that given a chance, every party will leave one seat between them and another party. How many people have entered a movie theater to find that they couldn’t sit with even a single party member unless they asked another party to move down? I propose that people should leave 2 or 4 seats between them and the nearest party.

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  25. how very strange… was just discussing this with some friends at the pub some time ago, although hadn’t bothered to do a formal analysis. would be interesting to do a genetic alg version based on heterogeneous urinal types (e.g., with dividers, differential spacing, etc..). then, I suppose the fitness evaluation fn would be the likelihood of someone else getting a peek at your goods (thus, things like increased distance or dividers would increase relative fitness).

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  26. According to my interpretation of the Protocol, pisser 3 in diagram 2 has violated the rules. He should have pissed in 3 or 5. If this was in a bar, pissers 4,5,6 or 7 are now allowed to piss on his leg as penalty.

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  27. for women… in toilets with a lot of stalls and not many people around the same applies: look for the one farthest apart from all the others. if there are three or less stalls or if there is a line in front of the toilet though the pack rate is 100% – since womens toilets have a tendency to have a waiting line in bars and restaurants and there are only stalls anyways all akwardness is left behind and efficiency is the most important factor.

    (that too is the main reason why girls in such places always take a friend with them: you don’t want to stand in line for 10 minutes either in akward silence or talking to some stranger)

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