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 
George:
You could just install 10 rows of urinals with 1 urinal in each row.
You would get a packing efficiency of 100% 😛
The thing is: how you can get the best packing efficiency for each row
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Fibonacci packing works better — Each new person chooses a urinal in the largest open interval, but at the golden ratio point instead of the midpoint. In fact, this strategy gives the optimal packing density. See Knuth I for details.
Of course, this requires a small constant factor of altruism on the part of the homophobic/germophobic/ureophobic participants.
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nil awkwardness occurs at frat parties when there’s one bathroom and a 30-minute line. 20-minutes in you’re best friends with your line neighbors and you’ve got a pact to get 3 people using one toilet and 2 in the sink. Now that’s efficiency!
Yeah, this also doesn’t take into account ballgames and concerts and such where you fill in every stall or you’re never gonna pee. But I think this relates back to lowered awkwardness levels.
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Check out “The Men’s Room” video at http://www.thepetersonproject.com. It’s a very humorous discussion of bathroom etiquette.
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I do not disagree with the math here, but the basic seems to be wrong.
this model would only work if a spesific amount of men were entering at the same time.
Different men with different bladders dont use the same amount of time to take a piss, and when you have started to piss, you cant change your place, youre stuck with the place you chosen in the first place. In a bar, ppl dont wait, if you have to piss, you piss. You cant predict how men are standing even if you know the number of urinals and men,
—
You enter a urinal, and there is one man standing in far left, then the alternatives would be (“O” is the already standing man, “-” is free space “X” is you)
OX— way to clingy
O-X– still a bit clingy
O–X- best choice
O—X homophobic
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This is SO gay. “Oh my, if I pee next to another man, it will be AWK-ward”. Real men can walk up to a trough, sidle up next to another fellow doing his business, and just whiz. No “shy bladder”, no blushing gussies.
Man up and just piss!
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This is not an international version of the protocol. It seems to be the “uptight person version”.
I suggest visiting an urinal on the Munich Oktoberfest where you could easily see packing ratios of 120% and more (even when presuming that 100% of the visitors are plain heteros).
Loosen up guys, there’s nothing to be ashamed of.
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Apparently there’s an app for that
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Hmm, this is quite true in the event that only stand up urinals are involved. I’ve noticed a variation though (dunno if this was an extension of the above mentioned). If in the same rest area there is a mixture of stand up urinals and full sized toilets, I’ve noticed that the most common pattern by far follows the above pattern for the first person, however, in an attempt to lesson awkwardness even more the second person will generally pick to use the full sized toilet stall instead (see below). This will continue as the stall empties and stand up urinals free/fill up.
I also noticed that the stall setup may have an effect. When there are small numbers of stalls, the stall’s size and layout may effect the decision making process. For example, if there are two stalls in a side-by-side layout, one is larger (the handicapped accessible one) but the doors themselves are flush then it feels like the stalls (distance wise at least) offer as much awkwardness protection if not more because of walls, spacing, depth inside the room, etc. However, in the case that there are two stalls and the increased size of the handicapped stall forces the placement of a door perpendicular to the other door, you could say it “feels” like there’s less space (because most of the time there is anyway but that’s besides the point) and I see people instead choose a stand up urinal at this point.
I do find it interesting though that I’m not the only one that has thought about this in such a way… Should there be an RFC for this?
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Great article man, but there may be a way to include people not entering the bathroom one at a time. In the second scenario, the one with 7 urinals, if number 3 and 4 enter at the same time they can use the 3rd and 5th urinal each maximizing the use of the urinals at almost 60%
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Here in Germany, the protocol seems to be more efficient. Instead of choosing the one with the most space to both sides, you just have to choose one with both direct neighbors free. This provides an efficient way to use 7-urinal rows and is muss less vulnerable. Furthermore, there don’t occur problems with different peeing times. Awesome! Awkward!
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Another rant:
Actually for n=4 this *is* optimal; also for n=6,8,10,16,18,32,34,64,66, …
Generally: for any even number of the form n = 2^k or n = 2^k+2, k>0, we get f(n) = n/2 exactly.
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Great application for small group awkwardness when someone gets real.
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Of course, this is also assuming that walls are not present, in which case it is very acceptable for each urinal to be occupied. Since said barriers cut off all sight and splash risks, men can be comfortable using every urinal, increasing efficiency to 100%, so let’s just put up the walls, people!
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I have also seen some urinals separated by some sort of small plywood or stained glass panel, with almost the same height as the urinal, so it blocks the angle of vision of other guys junk. This greatly reduces the awkward factor and allows for maximum use of the urinals.
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In my floor’s restroom there are only two urinals. 50% packing I suppose, but low threshold until Awkwardness.
What’s the awkwardness multiplier if the other dude is your advisor?
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I disagree pokerman.
O?X- best choice
O?X homophobic
In your ‘best choice’ scenario, if one more person enters who ignores etiquette, you are *guaranteed* to have someone beside you. Whereas with the homophobic option, there’s a 50% chance you won’t. Which might be homophobic, but is still best choice.
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Here’s another wrinkle – I’m tall, 195 cm, and I don’t want to use the lower urinals for little kids, which they often put on the ends, unless I have to. So I’ll take the open regular height one, which can disrupt the filling pattern or induce awkwardness. (I can also usually see over the divider if I wanted too.)
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http://www.addictinggames.com/theurinalgame.html
Back in the olden days, I had this game downloaded, but I see that it’s available remotely now.
Enjoy!
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One word: Dividers. Deep enough dividers cause 100% efficiency.
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Your solution remains inefficient. Here’s a better one:
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Randall, your clever analysis, while holding true for workplaces, supermarkets, restaurants and sports arenas, unfortunately does not hold true for bars.
As the current time T approaches closing time Tc, the number of available non-awkward sports A approaches K+1, where K is the result of the formula in your article. The simple reason is that drunk guys will piss in the sink.
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@JeffE — I think the point is we’re assuming that peers are unaware of packing schemes, and simply act in a generally predictable manner. The question isn’t how peers can maximize packing by conscious choice — it’s how a bathroom designer can use their predictable behavior to maximize packing invisibly.
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You have not taken into consideration sexual orientation considerations which could mitigate the validity of the basis of your mathmatical premise. Foolish Boy….
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im lucky i dont have to go through this. Not only can i hold my bladder for days butI cant even pee outsisde of my own home.
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Not sure if anyone else suggested this, but the comments I read through seem not to have. You neglected the best way to increase awkwardness, which is just choose the urinal directly next to the one guy there already. If you have, say, 7 urinals, and he’s at the very end, just pee in the 6th one. He’ll really wonder wtf is wrong with you.
Then, of course, if you want it to be really really awkward you can just try to share his urinal…
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If only most men followed protocol… How many times have I been in a block of urinals with plenty of free ones down the row only to have the next guy sidle up right beside me? If I had a nickel for every time that happened I’d have at least $1.85.
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International Choice of Urinal Protocol (ICUP). Nice.
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You could have saved the math and wrote “Urinal installers should put those metal dividers between the units.”
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Dividers. Very simple solution, leads to 100% efficiency.
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I propose that the following standard be used:
If all urinals are empty, choose an end urinal
If some urinals are taken, choose the closest non awkward urinal to the door that is an even number of urinals from the nearest man.
Addendum: if there is a large gap in the middle of the row, choose the second closest non-awkward urinal that is an even number of urinals from the nearest man.
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“Dividers. Very simple solution, leads to 100% efficiency.”
That’s exactly what I came here to say. But I will add that they be floor to ceiling. Nothing is more pointless or irritating than ones so short you can see over them, or so high off the floor that short people can see you junk under them.
Simple, cheap, and 100% efficient. But yes, nerding the math was fun.
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You’ve forgotten that by having dividers, it changes the entire game. Suddenly, people are more willing to pack into the next stall, but their willingness depends on the divider coverage… Full coverage – ie, divider reaches ABOVE head height so you can’t see the face of the guy next to you works better than partial coverage – ie, the only thing really being hidden is your junk. And of course, there’s always the wild cards – individuals with the urinary equivalent of aspergbers syndrome, who often try to start up awkward conversations while peeing in the very next stall, or those whose bladders are so full they’ll take any available stall and throw the whole equation off. Ah, the joys of bladder mathematics…
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Here is some earlier research on this topic, which I think takes into account some nuances of the problem that you may have neglected.
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I am uncomfortable with the use of the word “packing” when discussing subjects involving the male organ.
Does this make me homophobic?
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I’m pretty sure this is stolen from “The Alphabet of Manliness” by Maddox.
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Wow, I never knew peeing could have such descriptions.
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Now the part of this whole thing I find concerning, is this algorithm all but guarantees completely unused urinals. Do the owners need to start rotating urinals every so often to even wear?
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This fails to take into account the fact that the first stall is also (inevitably going to become) a urinal.
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There is one invention or modification that makes this problem moot…. if you put dividers (walls or part walls) up between the stalls then when the urinals fill up in the way as described, the guys who have to end up peeing beside someone else will almost always end up doing so since there is a feeling of semi-privacy.
As a side note, I noticed this phenomena way back in University but I never did put it down on paper in terms of a formula (probably couldn’t have anyway!)
I’m personally pretty secure with myself, so if faced with the problem of having to pee beside someone because it is the only stall available, then I will. I, of course, will always choose one that has at least one empty stall between me and the other guy if at all possible.
One other thing I also noticed in University…. the exception to this sacred urinal law also breaks down with the amount of alcohol consumed by the pee-ers involved.
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It is easier and 100% effective To go Urinal to stall to Urinal…
Y|S|Y|S|Y|S|Y
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I find it amusing that the list of quantities with optimum efficiency (3,5,9,17,etc) is listed in the Online Encyclopedia of Integer Sequences as a Pisot sequence.
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But you missed the one perk of standing next to each other — it makes it easier to multitask.
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Here’s one to ramp up the akwardness quotient. I was in a restraunt in Carnac, in Brittany, France. The toilets* were downstairs. The single unisex stall** was occupied. There was a urinal.
Tucked under the stairs
Just outside the unisex stall
By some sort of store cupboard door
Also beside: the beer kegs.
I did not investigate the plumbing, however I also ordered no further beer from that establishment.
At a free Bob Dylan gog I once went to on a beach in spain, they’d deployed, as well as the usual portaloos, moulded plastic urinal stands. These had three urinals in a circle around a central pillar, with each urinal having shoulder height wings that almost enveloped you. Completely outdoors, yet oddly not akward.
*incidentally, I never came across any of the formerly prevalent squat’n’drops on this trip
**in reality, cupboard with loo in it.
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What about the “Rosa Parks” syndrome? If there are 33 urinals and someone’s in the first one, I’m not going to walk all the way down to 33 to get a spot…especially if I gotta go.
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With girls, in a vacant restroom with nobody in there, it seems that the rule of thumb is similar to urinal use, only if a group of girls go together into a bathroom, they’ll usually use stalls next to each other. Girls don’t really talk to each other on the toilet either, unless they came in with a conversation. If a toilet is backed up, or not flushed we tend to back the crap out of there.
As far as going longer: Girl toilets seem to have on average only one, maybe two, more toilet than the Boy room. That combined with a lack of ability to quickly go pee (There’s no equivalent to urinals where you walk up and go piss quickly and then get out, so if everybody ahead of you is pooping, god speed, my friend.) adds to time. Also periods and the changing and disposal of uterine lining catchers, the fact that a lot of women’s trousers (especially dress/suit trousers) have way too many buttons (Seriously, wtf is up with that?), and you have to lower and raise those trousers every single time you go… it adds up.
Girls have sink issues, too, because that’s where the mirrors are, and generally where people hang out for their friend who’s in the bathroom.
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