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 
There are two sets of urinals on different trains. The first starts in Bath, heading to London at 125MPH. There are 7 people wishing to use the 3 urinals.
The second train leaves London travelling to Bath at 94MPH. There are 114 people queuing to use the 7 urinals.
Each person takes on average 6 minutes to do their business before leaving.
How many people will have used the toilet from each train before the trains pass at the Boris the spider climbing frame, just outside Slough?
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Ethelthrfrog – I bet your mum f***ing loved you coming home for holidays!
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For large values of n, men won’t be able to find the optimal urinal precisely (unless they stand there and count); they will merely choose a urinal which SEEMS optimal.
For example, if n=51 and the 3rd man is trying to find the middle urinal, he will merely opt for one which is “near enough” to the centre.
Therefore the urinal which he chooses is a probability distribution, presumably Gaussian, centred around the urinal which is in fact optimal.
I would like to see a fully Bayesian treatment of this protocol before I would even consider publishing.
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Oh right, Thailand has woken up, has she?
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Why isn’t urinal load considered when evaluating the performance of the protocol?
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Does anyone know where Mikey is today?
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For those unfamiliar with the urinal protocol, you may find this public service announcement informative:
http://z-studios.com/films/mre/
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In civilized countries (India), we have partitions between urinals, thereby voiding the necessity to go farthest from an existing pee-er.
That said, my algorithm is more dependent on the stride length and turn-radius into the urinal.
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Most of the toilets that I’ve used have gone the sensible route and just built more stalls. The urinals are mostly just there as an emergency measure I think.
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Partitions are nice too; what would be best (if not most efficient) would be stalls alternating with urinals.
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Okay, I’m a chick, but I think this is stupid. They should leave space between all the urinals about the width of one urinal apart, and put little partitions on the sides of the urinals, just about waist high. Then no urinals will be wasted. 🙂
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I have to echo Anonymous, there. If the supposedly-optimal urinal for me is clogged, or used but not yet flushed, I’m much more likely to make use of a different, potentially more awkward urinal, as I feel less awkward peeing next to another guy than I would getting someone’s pre-disposed urine splashing back at me.
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Love it. You’ve put “civilised”, and then next to it you’ve put “India”
LMAO
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Can I get a little analysis on the continuous case– the ever-classy urinal /trough filled with ice?
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I saw a poo in a urinal once.
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I find (here in the UK) on trains, and in fact anywhere there’s public seating, the rule’s slightly different.
While person B obviously can’t sit close to person A – that would be invasive (we have a culture of negative politeness in the UK – it’s polite to avoid bothering people – compared to the US’s culture of positive politeness – it’s polite to include people. This is why English people ignore each other so determinedly on public transport – it’s not unfriendliness, it’s actually politeness.).
However, they equally can’t sit as far away from that person as possible – that would be rude, it would suggest that they found the person objectionable in some way.
So, person A goes wherever they like. Person B sits roughly halfway between person A and the end of the row that’s furthest from person A. Person C then picks the biggest remaining gap and sits roughly halfway along it, and so on.
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Was it floating?
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No. Some small children were poking it with a stick. I think it was dead.
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Now, how about urinal rows that wrap around corners? The line of sight awkwardness no longer applies at the corner units, as you are facing in different directions. They do, however, increase the “accidental brush up” awkwardness quotient significantly (especially when you consider it’s a possible butt-to-butt brush up too).
Should they be treated as two rows, or one? Or should the corner units just be treated as one unit?
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I’ve developed a new method for using a urinal which doesn’t require you to be so close to the unit. I call it the “pee-over” manouvere. The only problem is that you need to be accurate, particularly in busy toilets.
As a woman, I generally find people shout at me when I demonstrate my technique.
Would anyone like me to draw them a picture?
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“Love it. You’ve put “civilised”, and then next to it you’ve put “India””
He also neglected that those partitions are just stuck on the side of buildings right on the street…Incredible indeed.
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Austin – are you telling me Indian urinals are just on the side of the building? So what do you do if you want to do a poo? Hold a blanket up?
I also find the drainage diameter of Indian urinals generally too small for the average European turd. . you end up having to ram it down there with a broom handle or other such object.
And that increases the risk of splashage too.
The man from Del-monte is most definitely not dressed to deal with poo splashing on the average Indian street.
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I think we’re all missing the real point of this post – that we need, nay, DEMAND a LaTeX font that renders the math environment in Randall’s handwriting.
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Once or twice I’ve seen a woman using those portable urinals in festivals, that must require some balls 🙂
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Can anyone tell me the angle of Elton Johns wig rotation today please?
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Further complicated when you add in an “Expert Urinal”:
http://www.urbandictionary.com/define.php?term=expert%20urinal
I once posted a “Urinal Challenge” where I give the restroom layout and which urinals are in use, and you try to guess which one I used:
http://jeff-simpson.livejournal.com/266073.html
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Partitions are where it’s at, and most bathrooms use them. Chest height is average. I follow the most basic packing algorithm, but it’s not really awkward with the partitions.
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Does the formula change for non-linearly arranged urinals? I saw one in Munich where the room had a thick divider wall splitting it into two U-shaped areas of urinals.
Also, I loved the way Asimov extrapolated the differing behaviours of the sexes in Caves of Steel: in Cities, where everyone shares the communal bathroom, women regard their bathrooms as the center of their social world, while men neither talk nor make eye contact; to do so is to court social ostracism, and Bailey spends about ten minutes agonising over whether Daneel is going to embarrass him in there.
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So get this for least-optimal planning at work:
5 urinals, last one child size (C).
F F F F C
Small dividing wall, but everyone else avoids C, and apparently some engineers don’t even know this social rule and tend to use the one next to C first leaving only the first to be used without awkwardness.
5 Stalls, 2 handicapped which are also avoided
H H F F F
If the whole place is empty, and I sit down in the right most stall, without fail, the one next to me is the next to be occupied. People suck.
captcha: squealer Episcopal
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The pee-over manoeuvre is fairly discreet in women’s toilets however. If done right, they won’t even know you were there. And by done right I mean crouching with each foot on the top edges of the cubical walls and aiming perfectly for the gap between the back edge of the toilet seat and the current occupant’s back. The more experience playing space invaders the better.
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Has anyone factored in splash back?
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Simon – you shouldn’t have to. Not with this new design. Looks quite fun, no? But useless for number 2’s. Unless you’ve just eaten a spicy curry!
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I’m not gonna check your math, but 4 urinals is efficient, not inefficient.
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Hello from Reddit!
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What if you have one long trough?
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I always wondered why when there is a lineup for the bathroom, there is always a lineup for the women’s bathroom, but there usually is no lineup for the men’s bathroom. So it feels like the empty urinals or toilets in the men’s washroom are being ‘wasted’ and the money should have been spent on creating an extra toilet in the women’s bathroom.
When public washrooms/restrooms are being built, how many toilets/urinals should they assign to each gendered washroom to maximize efficiency? However, this is harder to analyze mathematically, since you would have to consider the price difference between toilet stalls and urinals, and you would have to figure out the average time it takes for a woman to use the washroom/restroom relative to the time it takes for a man.
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So, what’s the protocol for those trough style urinals you see in ball parks sometimes?
I’ve always either waited for a stall (not recommended unless the janitorial staff is top notch) or held it in until it causes bladder failure (also not recommended).
Interesting note: I searched “Troth urinal” in Google, so I could figure out how to spell it, and the first hit was a forum topic where the members were asking where to find trough urinals in their area. It was a little creepy.
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Grant – ‘course you were. Accidental, eh? [wink wink]
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I wouldn’t call this “international.” It seems like whenever I’m in Europe people are always choosing a urinal too-close.
Maybe the “American National Choice of Urinal Protocol”
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Well how about urinals without rooms/doors? Here is the Swedish way. It is great for festivals and stuff. Plus, if you are awesomely tall you can stand face to face with another dude and piss.. ehm..
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I think the equations need to take into account that the awkwardness level is variable. If the bathroom in question is in a bar and everyone is 3 sheets to the wind on a Friday night then the awkwardness is next to nil (nil would be scary, that would be people sharing urinals).
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You forgot to put in for maximum awkwardness the variable of dropping your droors past your ass. Which will screw up the whole equation because then everyone avoids you at all costs.
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I would like to point out that, in the formula for finding the worst-case-scenario urinal-counts, the division by two is superfluous:
3*2^k/2 = 3*2^j, where j = k-1.
On another note, it becomes even more awkward if k<0.
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What about relativistic considerations, Ian?
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Also, did anyone notice what happens if you make an acronym out of International Choice of Urinal Protocol and read it aloud?
Or did everyone, and I’m a dunce for thinking that was subtle?
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How about this: install two rows of urinals on opposite walls with 5 urinals in each row. You get a packing efficiency of 60% for a larger number of people. That’s better than the 50% peaks in your graph.
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There are only 2 in the bathroom at work, you either get one of those when empty or go to one of the 2 stalls.
Great submission, Randall.
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