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 
At a yankee game the last one in uses the sink.
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It took me so long to work it out that I had a little accident.
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We’re dealing with maximizing urinal efficiency in a restroom where each guy comes in one at a time (though without any time between them) and pees indefinitely.
But, what happens to the construction if we were to allot a time constraint between each new guy entering and the amount of time each guy has to urinate, and change the goal from efficient restroom occupancy to an efficient level of service (arguably the better goal of restroom design)? If we were to use the 5 urinal set-up, for example, and give each guy a minute to pee and 15 seconds between each new guy coming in, guy #4 comes into a state of awkwardness where he is unwilling to pee and ends up peeing himself (thus resulting in awkwardness anyway). Plus every new guy thereafter.
However, an 8 urinal set-up, given these time constraints, would allow each guy to come in as another leaves without any akwardness, neither from standing next to a peeing dude nor dancing while waiting for a urinal to free up. 8 becomes much better than 5.
In my mind, the ideal bathroom is one in which there is no waiting or awkwardness. So my question is: what do you think are the maximum amount of time allotted for peeing, the minimum amount of time allotted for new guys to enter the restroom, and, therefore, the more practical optimal number of urinals in super-busy-restroom?
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My problem with all these discussions is nobody ever brings up the issue of child urinals. It is obvious that no one wants to use the small urinal, but these systems never factor that.
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A female specific example may be found in the “trying to conceive” community. These women spend hours building and analysing charts and graphs. Generally, women are only fertile for three days a month, and often these fertile windows can only be predicted with certainty 24 hours in advance.
http://www.fertilityfriend.com/videos.html
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“”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?””
Wait………..is the person behind XKCD a guy? Holy Mother of God!!! I always thought it was a woman…..
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Heh..I didn’t know that guys have this kind of problem. Mines worse
In my school there is only 20 toilets to facilitate 900 girls so you only go to the toilet when you have to because they are so disgusting. To make it worse, there are toilets notorious for leaking, no lock doors and lack of toilet paper and unmentionables in the bowl. So the protocol for us is if you are first in line you check for the best one and use, and the next person does the same. When all the avaliable toilets are taken then you enter the toilet that someone has came out of because if they have used it with an okay face then it’s safe to use.
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As far as the “Eyes Forward at All Times” convention is concerned, there will be no difference between 8 urinals (without “Eyes Forward at All Times” convention) and 4 urinals (with “Eyes Forward at All Times” convention). So, if you have room for 8 urinals, just put 4 urinals instead and utilize the remaining area for some other insteresting stuff (vending machines etc.).
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Well,the urinals could be placed at a reasonable distance from each other.. this way there would be no problem occupying any available urinal. Theory of ” Being least conscious” could be applicable in this case.
One more possibility could be to bring forward the buffer urinals..
Illustration as follows:-
—————– Wall
O O O O
O O O
[O = urinal]
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You probably know this one:
http://www.research.att.com/~njas/sequences/?q=3%2C5%2C9%2C17%2C33&sort=0&fmt=0&language=english&go=Search
Cheers.
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Why don’t they save money on plumbing and urinal units by spacing them widely?
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TomO says:
September 9, 2009 at 1:43 am
My problem with all these discussions is nobody ever brings up the issue of child urinals. It is obvious that no one wants to use the small urinal, but these systems never factor that.
ANSWER
The child urinal really poses no problem. Just start peeing and turn to the guy next to you and say “if your dick were long enough you could pee here, too.”
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observe: the two expressions for the 50% and the 33% teh difference between two consequetive nos of the respective expressions (3-4=1, 5-7=2, 9-13=4 and so on it doubles 1,2,4….).
also i am very intrested to know how you derived the expression, the ‘int’ is integer right? couse if it is not then you would end up peeing in the middle–between urinals
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The study considers all Awkwardness as same, which is generally not the case. Preoccupants of the urinals affects the awkwardness of their adjacent slots. Lets take the case of 5 urinals assuming slots 1, 3 and 5 are taken for three gentlemen A,B and C with their waist in ratio 3:3:2.
1 2 3 4 5
A B C
1.5x 1.5x x
So from reference point of a new occupant slot 4 will have the least degree of awkwardness.
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well hemanshu, this is what just happened:
i go to your blog site and check out that bike image, i click on that image and learn it was clicked by siddhart menon. now i know siddhart menon through a frien named ashlesh pol, and this was confirmend when i went through his flickr pics and foung my frien ashlesh in one of the pictures named ‘bored’.
1-XKCD
2-Hemanshu
3-Your Blog
4-I click on that bike pic
5-I find siddhart on it
6-Confirmation obtained through one of the pics where i just found one my friends
6 degrees of seperation.
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With women we have the stall choice. For instance, most women will not choose the first stall ( the one closest to the door), but will choose the second, then choose every other stall…. Am I right?
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As a female, the only thing I really care about when it comes to stalls is if the toilet is filled with poop or not. In high school, though, I’d purposefully not flush the toilet, only to come back in later classes to find out if someone else bothered flushing it.
Did that count as a valid social experiment?
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consider an alternate conundrum, the cardio equipment at the gym.
is there an optimal number there, assuming you need atleast 8-10 machines to offer satisfactory service to your patrons. more optional.
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I figured out the equivalent for us ladies today! Sorta.
In a class room, girls will sit together only if they know each other. If not, there’s a unit (table, desk, chair, whatever) in between one girl and the next until someone voids this unwritten rules, at which point we all just start sitting next to each other.
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What if there’s two rows facing each other closely?
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Oddly enough, someone actually did research on this, and the protocol is more like any newcomer will choose the largest gap and halve it. Thats certaing what I do. Empty urinal, stand in the middle (unless it’s a 3 seater, in which case one end is chosen).
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Or you could put partitions between the urinals.
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I am totally secure, and not awkward.
Not only will I pull up in the urinal next to you, I’ll start up a chat.
See you in the WC!
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You guys have way too much time on your hands!!
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This is a total and complete rip-off of Maddox’s alphabet of manliness. You’re a shameless plagariazer
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It would be interesting to see the best/worst days to have a period, due to the days covered by subsequent periods (based on an average 28-day cycle and 5-day period). For instance, any period covering November 27 may also cover Christmas. December 4th is no good either because of New Year’s Eve. I’m wondering if a women on birth control pills has figured this out (with all the other holidays) & changed her start date accordingly…
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You could always put up large dividers between the urinals. Either that, or place urinals on both sides of the restroom. There is also another interesting setup: Separate the urinals with bathroom stalls, garbage cans, or anything else one might find in such a room. The awkwardness can be relieved, and urinal layouts can be optimized for any desired level of occupancy.
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it’s really true: when a group of women work together, they eventually start cycling their periods at the same time. it’s a fact, where it came from i dunno, but i know there was a study done. it would be fun to see your math on the whole thing.
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What about the effects of kiddie urinals? When one is at the end, the first guy naturally takes the non-kiddie end urinal, which then leaves the next one with a quandry: to minimize awkwardness, they should be at the other end, but that would require using the kiddie urinal, which generates its own awkwardness. And, of course, one doesn’t really want to use either it *or* the one next to it, because if a child comes in, you’d then be next to the child, which is even more awkward than being next to another adult.
We really need a controlled study on this.
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I like to pee on the guy’s shoe next to me… and then say, “Sorry, I thought you were someone else.”
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As a woman working in commercial real estate, I can tell you that my company installs urinal partitions in the bathrooms in all the high-end office buildings we run. I can’t tell you if it has any change on the Protocol though. If it doesn’t, maybe we should save our money on the partitions and just install the optimum number of urinals in the first place.
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Unfortunately the formula doesn’t hold for places with horrible bathrooms like the Minnesota State Fair, where the conventional row of urinals has been replaced with three “pee-troughs,” which have no side walls, are 5-6 feet wide, and stand roughly 2 and a half feet high.
The protocol, which is presumably designed to prevent bathroom users from getting an eyeful of mansausage, pretty much breaks down in these situations. Awkwardness is had all around, as men try to pee as quickly as possible and get out of the bathroom without having to speak to or acknowledge any of the other men whose genitals have been on semi-public display.
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I agree protocol is thrown out the window at county and state fairs. I am a carny and have seen some of the worst troth style community urinals and over flowing porta-johns in existence. Many times walking out and holding it for a couple of hours is much more pleasant, or simply finding some privacy behind a horse barn to take a leak. Human shit is worse than any bullshit.
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If we could somehow quantify the desirability of clothing designs based on a few (dozen (dozen)) variables, we could probably figure out a way to avoid (or intentionally provoke) the “we’re wearing the same X today” awkwardness that invariably occurs within any gathering of women.
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I shared this with my BFF who is getting his PhD in Combinatorics. This is what he had to say:
I think it just might be the most pressing combinatorial issue of our times. Though it doesn’t make mention of several critical variables, like dividers, or attendants, or level of drunkenness and thus tolerance for Awkwardness.
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I was always taught that you could model this simply with Hund’s Rule.
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I’m glad to see that others have thought about this issue as well. Unfortunately, the efficacy of your protocol has too many situational assumptions to functional effectively. There are certain groups of people who are very likely to undermine the basic premise. The dumb and intoxicated are oblivious and perhaps apathetic towards the Awkwardness and their choice of urinal is normally random. Creeps will maximize Awkwardness and choose urinals closer to others urinating. Gay men with sexual tension between each other may also choose urinals closer to each other.
My solution: pee in the stall.
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I agree with most of the stall protocol that some of the other women mentioned above – taking every other stall and avoiding the one closest to the door, also waiting for other women when there’s only one sink.
However, often we go to the bathroom with a friend, while having a conversation. When that happens we usually choose the stalls next to each other.. and continue the conversation. When there are no two adjacent available stalls, we pick whichever ones are available and continue the conversation, over the other woman.. and every so often they may join in the conversation lol. That only happens though when we all know each other.
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Easy way for making them pee without having to choose: Put a bigger distance between the urinals and save money from buying more urinals than your algorithm proposes! So simple…
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Just dropping in to say that Melissa’s comment made me crack up at work. It was the “smelling of altoids and lip gloss” comment, cos seriously, we’ve all been there.
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By selecting a sufficiently large number of urinals, the problem will be eliminated, and will also lead you to being the Pee King – not to be confused with Peiking, which is in China. That’s too far to go to pee, but there are lots of trees on the way.
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A useful tip to any owner of an establishment with an ineffective number of urinals who wants to get maximum nonawkward use is to place an out of order sign on every other urinal starting with the second urinal.
this will ensure at least 50% or greater utilization.
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I don’t know of any girl-specific examples, but I think this sort of thing happens any time you are with a large group of strangers. For example, in a lecture hall, a computer lab, a train, or a movie theater.
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