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 
I’m an Australian, troughs used to be the norm, they’re still common.
People spread out much as Randall describes, but I don’t consider it awkward until you have to jostle for a position, and some drunk pisses on you.
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Boom. Blown out of the urinals.
The trough – if only the architectural world would wake up to the trough. Such a paradigm shift would make the world a better place and could put our arguments about urinal efficiency firmly in the past.
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The standard way to bypass the whole ICUP issue is small barriers sticking aproximately 1 foot out from the wall between the urinals. That distance is enough to block ‘accidental’ viewing and guys will no longer hesitate to use a urinal adjacent to one already in use.
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I discovered recently that Awkwardness can be mitigated, and mitigated especially well when done by accident.
I’d been singing to my son earlier that day, and had “The wheels on the bus” going round in my head.
On entering the gents, there was only one urinal free. Awkwardness. Then I found myself whistling…. and the chap next to me said “The wheels on the bus, eh?”
To which there’s only one reply: “Yeh mate. Go round and round, they do”
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With one long urinal you normally end up with the same spacing – except the guys on the end will pee into the corner and people will often try to avoid the going in the middle.
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What about bathrooms with urinal partitions?
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I don’t find anything awkward. I keep my eyes off of everyone’s dick… and I really don’t give two shits if they look at mine.
I caught a guy looking at my dick once and after I finished peeing I turned to him while clenching my rod and I smiled, asking him if he’d like to suck it too.
Creeped him the fuck out. Win.
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Even more awkward is a trough urinal placed in the middle of the mens room floor leading to potential cross streaming, and looking forward at another dudes junk or making eye contact.
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awkwardness arising from a biological and necessary function is pathetic.
The trough design is the NATURAL best choice- talk about efficient.
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What is the base of the logarithm? I’m assuming that’s what the “lg(” is.
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There’s a way to get 100% efficiency. Dividers!! whenever I walk into a bathroom I always hope there are dividers between the urinals. It they take away the awkwardness.
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Revised ICUP:
Guy 1 will choose leftmost urinal that is not a ‘child’ urinal.
Guy n+1 will use whatever non-‘child’ urinal maximizes his distance from all other users if one is available that does not cause awkwardness. If no non-awkwardness inducing, non-‘child’ urinal is available, he will choose whichever ‘child’ urinal maximizes distance from other users (and will refrain if no such urinal exists).
(in other words, each guy will take a slightly closer adult urinal over a better placed child urinal, as long as such choice does not induce awkwardness).
Theorem: For any positive integer n, there is ‘child’ urinal/non-‘child’ urinal distribution of length n that forces maximum efficiency.
Proof: Counting from left, let the odd numbered urinals be non-‘child’, the even numbered urinals are ‘child’. (note that this brute force method requires either (n-1)/2 or n/2 ‘child’ urinals, depending on whether n is odd or even)
Open Question: Given a positive integer n, what is the minimum number of ‘child’ urinals that forces maximum efficiency?
Partial Result: If n = 2^k + 1 (for some k), then the answer is 0.
Partial Result: If n = 7, then the answer is 1. (Much better than the brute force method result of 3!)
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Western Earth Choice of Urinal Protocol works as a backronym just fine, and is slightly more accurate.
Appropriate Captcha time: Unacceptable Voodoo
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Another factor unmentioned is the awkward child urinal. Does the child urinal count for n or is it omitted?
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You can fit 4 people on 7 urinals if each person is just one urinal away from the other. Instead of choosing the middle one.
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The solution is very simple. An LCD screen that shows lingerie models at each urinal.
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Great post! I’ve been meaning to write on this very topic for years but always thought of it when I was in the situation, then quickly put it out of my mind, which is typical of my public bathroom experiences as a whole.
My theory is essentially the same, but I had not thought to develop an accompanying mathematical formula. Well done!
Regarding the idea that this is somehow homophobic… you just don’t get it. It’s really just self-consciousness, and I think perhaps a lower-level instinctive urge for privacy and seclusion, for the sake of safety: you’re extremely vulnerable at these times, so you want to protect yourself by being as far away from others as possible.
Or maybe not. But I challenge anyone to say it wouldn’t feel awkward if you were alone at the end urinal and someone came in and used the one right next to you instead of leaving at least one space between you.
Regarding the trough: ugh… it’s the worst possible solution, like an infinite row of urinals with zero distance between them.
Anyway, it’s reassuring to know I’m not alone in disliking urinals and wanting to place myself as far as possible from others when using one.
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Jack, it’s not about being homophobic. It’s about other things, such as splash. Nothing like having another guys urine getting on your pants, or in some cases on your feet while you’re wearing flip flops. There isn’t just splash from the urine hitting the urinal, but also from the flush. If the guy next to you flushes and walks away, the water-urine mixture has the chance, depending on the urinal design, to splash you too. Walls, do help minimize this, but that means the owner of the bathroom needs to put money into something with absolutely no return.
The trough is always an interesting idea. Nothing like feeling ashamed/proud while peeing in front of a bunch of guys.
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How does your equation change if there is a mix of gays in the urinals!? 🙂
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This is really fun to apply to riding the subway!
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It’s purely homophobia. Sad for you.
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Would dividers between urinals, sufficient to prevent “comparison curiosity”, affect the formula by reducing akwardness. Also does the placement of the queue (do the men waiting for a stall wait near the middle of the stalls or near one particular end) affect the area they will initially approach as least akward.
Ah forget it, I gotta pee.
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There is also the event where simultaneous exits can affect the decision making. In a 5 stall bathroom where stalls 1 and 3 leave simultaneously (thus having the flush and recent “aroma” lingering) the next client may well take stall 2 (a “fresher” location) and disrupt the efficiency.
Ah forget it, I gotta pee.
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Since the article asked for some feminine practices that should be analyzed, how about this: It is a known fact that women MUST go to the bathrooms in groups. We will NEVER know why, it just is, kind of like gravity, it just is. So in varying sizes of groups of women, what is the acceptable number per group to leve at one time to use the “powder room”?
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In my experience, the protocol can allow for higher efficiency than you’ve show. It’s a more advanced version of the protocol perhaps, and depends on how badly you have to pee and whether people are lining up behind you.
With 8 urinals, you’re essentially correct about the order of the initial four, though I find the protocol usually has the very first person choosing the urinal furthest from the door, which in this scenario, will be to left of the urinals. So, the first four would go: 8, 1, 4, 6
Ideally, this would be the total as in your scenario, but if you must go or there’s a line forming it moves to the next level. The next urinal you should fill is number 5. At first this might seem more awkward than 2 or 3 because it’s between two people, but it avoids “pairing in isolation” and so is actually less awkward. It is better than the same scenario in number 7, which is also between two people, because for number 7 you would have to travel further and so it appears like you selected the people to join rather simply optimizing the convenience to awkwardness ratio.
The next one to fill is debatable. The philosophy of the previous selection might dictate urinal number 7, but number 3 is now simply the end of a line of 3 other people so you no longer need to worry about “pairing in isolation”. It is also a shorter distance to travel and leaves you a person on only one side, which is preferable to a person on both sides, so number 3 is almost certainly the better choice.
The next choice is urinal number 2. Both 2 and 7 now have people on either side, but 2 is the shorter distance from you.
7 fills up last.
So the order is 8, 1, 4, 6, 5, 3, 2, 7 and 100% efficiency is achieved with the least amount of awkwardness. Ideally you could limit the number of people to four, but it’s conceivable that you might find yourself in a scenario where 8 or more people arrive at the washroom at the same time and a line begins to form. If so, this is the way to fill up the slots.
With 7 slots, the ideal scenario is 7, 1, 4, but if this happens the next person is kinda screwed as the “pairing in isolation” is unavoidable. If the third person is considerate they should choose number 3 instead of 4. Then the order would go 7, 1, 3, 5, 2, 4, 6 and you again have 100% efficiency with least amount of awkwardness.
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What happens when you factor in the mini urinal that almost every bathroom has? If you walk in and there are 3 urinals, 2 large and one small with the small one at either end, and someone is at the large end urinal, you are faced with a decision: either pee in the small urninal, which is awkward in itself, or take the middle large urinal which also creates awkwardness. Should the architects not install the small urinal between the large ones for the few times a child needs to pee in a urinal? The majority of the time they are led by their parents to a toilet to pee.
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I’m glad I’m a girl… poor males, why can’t you just pee in nicely boxed private toilets like most of us girls do? I get ‘stage-fright’ if any other chicks could possibly hear me fart… I can’t imagine how awkward it must be to pull it out and have to pee in plain sight of all the other boys in the room, including ‘Large Tool’… ugh…
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From my experience in the UK, those long-trough urinals are probably the most efficient as everyone using them sort of just accepts that there will be awkwardness and gets on with it. The no crossing streams rule still applies, but because there are no physical indicators of exact spaces to line up with, the results tend to be men closer together. However, one usually finds that they still start at either end and gradually fill in the intervening space until the proximity has reaqched a maximum level of awkwardness, at which point the others just have to wait until a space frees up – which is then made more awkward since as soon as one bloke leaves the trough, the guys either side will inevitably shuffle across to finish off in the space he was occupying to avoid remaining in awkwardness with those on their other side. So if you’re waiting for a trough and a space opens up, either be quick or wait for awkwardness to drop, which could take a while if the queue behind you is full of men too impatient and of the ‘be quick’ mindset to wait for you to go. You snooze, you lose. You get awkward easily, you lose.
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This is hilarious, although I do feel that this could apply to a woman’s restroom as well. No one wants to sit in the stall next to someone either. And then if you factor in the mathematical probability that one or more of the stalls might be being used as shitters, that complicates the equation.
For men, at least you know 99% of the stalls are being used for shitting with the other 1% for shy people, coke users or gay sex.
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I agree with Large Tool i do the same thing and i’m not gay
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I must admit the awkwardness is not problem for me, the children urinal is especially well suited for those that are well endowed. Seem to be made for us!!!
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i say you are all weird. who cares where you stand! if there are 5 urinals, might as well us them all. geez. be real men and just whip it out. so what if another dude sees ur dick. we’ve all got em.
and if you guys are worried about someone looking at ur dick, dont be. if someone looks, consider it a compliment that they would want to look.
no, but really, grow up and be real men. there are too many self-conscious people nowadays. just whip your dick out, do your business, and be on your way. thats what i say.
and, for the record, im all for trough urinals (more people can pee and people dont have to wait or hold up the stalls for people who actually have to take a shit).
so, grow up, be men, and stop worrying about people seeing you penis.
that is all
thank you
(by the way, i dont intend to down anyone else’s views, this is just what my opinion is)
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what about those washrooms with 2 urinals and 8 stalls, if someone is using a
urinal and most of the stalls are open I consider a stall an option
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How about mounting urinals on pillars rather than walls..
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I think a better protocol would be “the first guy picks an end urinal, and every subsequent guy chooses the urinal which is 1 urinal away from the last person entered”.
The minimum efficiency in this case would be 50% (for even no. of urinals) and for odd no. of urinals efficiency will be more than 50%.
As for the basic premise that “the first guy picks an end urinal, and every subsequent guy chooses the urinal which puts him furthest from anyone else peeing”, I think the marginal utility of increasing distance by one more urinal is very less as long as 1 urinal distance is maintained with every one else.
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I think it’s also important to keep in mind that nobody is going to walk down a line of two-hundred urinals because the first one was taken.
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Just to throw another spin on this. I use no urinals. I always go into a cubicle and lock the door.
Thoughts please?
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Well, if you DID see akwardness in a public restroom and no way to avoid it, then CHEAT! Go into a stall!
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To append my previous statement, if partitions help to reduce Awkwardness, then why the hell did my school do away with the 3′-off-the-wall partitions in the middle of last year? Don’t they know that’s dangerous to the flow of the piss chain?!?
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This is really fun to apply to riding the subway!
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You can wonder at why guys are so homophobic. Weird.
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It seems to be nice post!! keep it up some good work!!
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What awkwardness? I always take the one that allows for a space between other goers, but if that option isn;t available and I got to piss, guess what? I piss in the open urinal and move on. Any awkwardness is all in your mind.
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