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.
xkcd 
This is interesting, but for me it fails to address the two fundamental opposing forces. Perhaps this is a very British point of view, but I think all the talk of splashback is really trying to disguise the real issues at work.
Your choice of urinal is immediately judged by the other men, to whom you cannot speak because they are strangers with their willies out. Your self esteem depends crucially on their judgement of you, and you will have to stand in silence with their judgement weighing down on you for the length of your overlapping urination time. This is why you feel the two competing pressures:
1. If you go too close to another man, everyone will think that you want to look at his willy. Wanting to look at willies is not a problem; wanting to look at them while the owner is using it to wee, and hasn’t given permission, is not. It makes you weird and creepy.
2. If you go too far away from other men, everyone will think that you are afraid of the other men seeing your willy, which can only be because it is very small or badly deformed. This makes you less of a man. We’ve all seen it: the guy right in the corner, turned slightly away for good measure, just in case someone sees his miniscule willy.
Hence guy number 2 does not go to the end one, he goes for a balance, not necessarily half way between.
Any thoughts on this theory!?
G
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I’d use html if I knew it, but I don’t. So copy & paste the link to see an example of a setup with a 100% usage rate, and 0% awkwardness.
http://viewmorepics.myspace.com/index.cfm?fuseaction=viewImage&friendID=1741182&albumID=420371&imageID=7760861
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never mind that html part, had no idea it did it for me.
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By the way, Amsterdam has the coolest public urinals: http://citynoise.org/article/566
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The only problem is, you forgot to variables (very important ones in fact).
1.) Stalls
Always a viable option, and important to avoid Awkwardness
2.) The Door
Sometimes the best choice is not to play…
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Man’s Man – You’re the guy we all try to avoid. You ARE that guy that groans when he wizzes and pees at the pressure of a fire hose splashing out on the people beside him. Face it buddy, the entire discussion is about you.
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Maybe they should just start putting individual urinals with big empty spaces between them, so they don’t waste urinals and everyone can pee at once.
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Well the Urinal Protocol isn’t really that simple, seeing as how there are multiple factors that negate the usual “one urinal buffer zone”. For example if there are walls between urinals it is generally not too awkward to pee next to another dude, if all other optimal choices are taken. Also if there is a “kiddy urinal” which is drastically lower on one side of the row, then the second person to pee will take the urinal next to that one. Also if there is a sink next to a urinal without a wall that urinal is rendered useless. Then there are other environments which have whole other sets of rules, such as high school bathrooms, office bathrooms, or even movie theater bathrooms.
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@Colin – In Australia we have the metal trough, but just as commonly we have urinals being discussed here too. General rule, if America and the UK do things differently, we’ll have both.
I’d like to add in another variable that will affect the protocol when reaching larger number of urinals. This can be described as the “far enough” rule, whereby if #1 is the first on the left and taken and there are a large number of urinals (say >= 10). Then walking ALL the way to the other end for #2 can be wasted effort when the urinator (or urinee?) feels comfortable with a buffer of 6 urinals, meaning they don’t take the end one but rather just the closest but far away enough urinal. This creates a break down in the usual protocol that spirals out of control until there is a period of time with no urinal users. I believe this would lead to greater inefficiency at higher urinal levels and so the higher efficiency levels stay with the lower urinal numbers.
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I have always thought the very idea of urinals pretty disgusting. But I had no idea there was such protocol involved. You’d better not be in too desparate a hurry if you’ve first got to work out where you should stand for optimum effiiency! Far enough away for no splashing I think would be my major concern. Thank goodness we women have our own individual rooms with, usually, lockable doors.
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I went to an all-male prep school, and the student government ran a campaign for “the 1-3-5 rule.” It encouraged people to use only the odd numbered urinals. This meant that you always used at least 50% of the available urinals (for even numbers- (k+1)/(2k+1) for odd).
It was stressed that you didn’t need to choose the third stall second, just that you should avoid even-numbered stalls.
As a former track runner, I have to say: the trough is a god send when you have 3 minutes to go pee and get to the starting line. I think I’ve only seen them in sports complexes.
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Great explanation!
A similar situation occurs with having to choose seats on a bus, or other small space where strangers must be in close proximity to each other. The bus problem is complicated by the fact that people sometimes arrive in groups that wish to sit together.
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WOW! I’m glad I found this site. of course this conundrum is totally annoying, especially when some guys are unaware. and by the way, there isn’t really an equivalent for women, seeing as they pee IN GROUPS! But you could find some other interesting to awkwardify girls 🙂
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This subject requires some field work. Theory is fine, but we need a man in the bush counting the actual behavior of the species. Any volunteers?
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This subject requires some field work. Theory is fine, but we need a man in the bush counting the actual behavior of the species in question. Any volunteers?
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I let my Girlfriend once play a game where you come into a bathroom and have to pick a urinal, she was actually quite good at it! Seems like girls would act the same, if they would have pee next to guys^^
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And almost every bar in san francisco.
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<<<>>>
And almost every bar in San Francisco.
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Solution that requires basic math skills (subtraction):
While making blueprints for restroom urinal placement, place 3-9 (or more) urinals in a row, then ELIMINATE the even ones all together.
You save money, and prevent awkwardness by having an invisible 3 foot barrier between your pouring faucet and the hose next urinal over.
Ex:
original plans:
UUUUUUU
new plans:
U U U U
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oops, missing a U
Also, one way to lower awkwardness is to place adequate barriers which prevent any eye-eye and eye-penis contact.
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lol this is too funny. truth is, most girls act the same way with the actual stalls in their bathrooms. so it’s not really gender specific. just not AS annoying when someone ends up in the next one over.
Most girls find it bad to have someone in the next stall because they feel like that person could be looking at their feet and they think it’s gross that they could actually see even that small part of them while they’re doing what they’re doing. *shrug* Not all girls feel this way, but I’m sure there are one or two guys that don’t follow this protocol either..
btw, guys, if you want to really piss off people just choose the urinal right next to whoever is already in there. Then ask them about their mom.
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This can all be solve by placing a separator wide and high enough between urinals. A little extra cost in the material can save lots of money in the long run of unused urinals and the frustration of waiting for the others to finish if every second urinal is used up.
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I’ll keep this in mind if I ever go into the bathroom construction business…
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we need more people like Randall Munroe in this world.
p.s. my Captcha was “ghetto cowhands.”
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… and mine was jonson 79. Just thought I’d add that.
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The solution for a 100% packing is to have partitions between urinals thats all…. 😛
nice work though!
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Hmm… interesting
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Boy, I stumbled here (Stumble Upon, I mean) and then I thought: hmm, interesting.
This is the sort of thing that is very likley interest the compulsive fellow physicist at xkcd. Maybe he would come up with a strip on it.
Before sending it, though, I watched the top of the page and only then I realised he had written it…
I will write 2000 times on the blackboard: “When I stumble I will watch the top of the page first”…
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Actually, the best way to make someone feel awkward at a urinal next to you is nonchalantly compliment them on their watch (or ring) after they’ve been peeing a bit.
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I peed in a horse once.
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hahahahaha at two sticks comment
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@Matt: A very elegant solution, but one that is too geared towards eliminating awkwardness at the expense of restroom efficiency. What you’re forgetting is that the even urinals serve a purpose: Although there is awkwardness, they can be peed in. If you have a 5-urinal array, they first fill up as Randal mentions. However, when guy #4 and #5 arrive, assuming the urinals are still occupied, they will not wait for an odd-numbered urinal to open up, they will use an even-numbered one and accept the awkwardness, particularly if there are additional guys in line behind them. Everyone knows about the awkwardness, but no one is going to bring it up as an actual reason to hold up a line if there are people waiting.
However, your solution does have merit. Rather than reducing the number of urinals from five to three, however, why not just add a three-foot buffer and increase the length of the restroom?
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Reminds me of this:
http://theflowfieldunity.com/2009/06/25/three-shakes/
The solution is alcohol.
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What a coincidence. I just happened to read an old paper on this very subject. From Uretics to Uremics: A Contribution toward the Ethnography of Peeing by Elliot Orring 1975. While Orring notes the obvious aversion to picking a urinal adjacent to a fellow pisser, it is not, in fact absolute, in crowded scenarios, restroom patrons are more likely to choose a urinal next to someone else than to wait for a vacancy, or use the stalls. Suboptimal urinal selection is awkward perhaps, but crippling to the urinal protocol, hardly.
However, I like the rigor with which you have approached this problem, so it’s due a closer analysis. The urinal protocol observed by Orring is in fact different from the one you have proposed. Orring studied the protocol in public restrooms using only arrangements of five urinals. With all urinals empty, urinal #1 of 5 (the one closest to the entrance) is, as you suspected the most frequently chosen. 3 and 5 are somewhat less likely, and 2 and 4 are the least likely of all to be chosen. This implies that restroom patrons have some sense of how to optimize their urinal selection, eschewing the even numbered stalls to allow greater than 50% non-awkward utilization. But so far so good, your model was never intended to account for things like out of order urinals, random deuces in the optimal urinals etc. Where your model comes into direct conflict with observed reality is when urinal 1 is occupied and a second patron enters the restroom. Pisser B as we’ll call him, is most likely to select urinal 3. This may because the requisite 1 urinal gap rule has been satisfied, and the second pisser feels no need to walk another 3 feet to #5, or because the second pisser wants to play it cool. Picking #5 might give the impression that pisser B is unduly afraid of someone else looking at his junk (maybe that would be interpreted as lack of confidence in the size of the member, or as a bit of undue homophobia), or it may give the impression that pisser B has trouble urinating in close proximity to another (also quite embarrassing for a man).
So from all this, we can gather two important facts. First, urinal protocol is not an issue of establishing maximum distance between pissers, but only of establishing the one urinal gap if possible. Second, urinal patrons are capable of choosing urinals based on knowledge of the protocols with the intent of minimizing awkwardness. So, in a seven urinal example, if the two end urinals are taken, the third pisser may well select urinal #3 or 5 rather than the strict middle choice of 4 in the interests of allowing one more patron to select a urinal without producing awkwardness. However, your analysis should be good for sowing awkwardness, but if you want to do that, just choose the urinal next to someone, and start singing.
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This is all well and good, but you seem to forget the problem of the short urinal that is on one end or the other. Even when all others are taken, only the most desperate take this option.
This doesn’t seem to have been noted, and I don’t have time to go through the other 500+ comments to see if anyone else brought this up, but I feel it is a valid point.
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Paul bring up the trough urinal, and it occurs to me: has anyone done any work on extending this problem to the continuum?
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Lol, I’ve known about this for some time now, yet a good deal of people STILL choose the even numbered stalls. Then again, it really doesn’t bother me where people wanna go pee, yet for others it seems to be a big deal.
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This whole question is silly. Install only two urinals and the “problem” is solved, save in the shameful minds of American men who seem to have no compunction about hunting and killing birds and animals, or humans, in the twin cases of Iraq and Afghanistan, but the notion that someone may actually catch a glimpse of their willy and they are reduced to nervous, fearful, quivering jelly. American men tend to be wimps. Is it American women who do this to their children?
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FYI to Rood. It is not a man’s willy that other men fear. It is failure to perform — e.g. pee — that is man’s greatest fear.
The truth has been spoken. Anonomouse strikes again!
Keep your distance gentlemen.
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An alternative solution – instead of having the urinals and washbasins segregated, replace every even-numbered urinal with a basin. Voila! Eliminate proximity-based awkwardness and queues for the hand-washing facilities at one stroke!
PS My Captcha is durex promise. Does anyone else find that weird?
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Does the age of the “participants” need to be accounted for and how about accounting for learned experiences from childhood? For example, I grew up in a home with 8 boys in a two bedroom house with one bathroom. The boys learned it was possible to fit three “participants” around the toilet at one time – and never miss…mom might disagree, but he point is that the awkward factor may be different for some “participants”.
And what of age – the young don’t care yet if someone may catch a glimpse and the old don’t give a rip any longer.
I’ve always preferred the stahls where I can make bubbles!
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Posted only for capcha: hollower life
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its only gay if you make it gay
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The major annoyance for women (also, scariness for men) when our periods sync up with each others’.
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