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 
What happens if one of the urinals is one of those lower ones meant for kids? I mean, I’m short enough that I can use the smaller ones without much splashback and in and office building it shouldn’t be a big deal but what about at Target or something? What protocol is more important, this one or the one where you leave the lower urinal just in case a kid shows up?
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I actually did my Westinghouse paper on this subject (actually, seating on a subway bench – somewhat less gross). But I was young and stupid and probably got all the math wrong.
I always wondered how it would generalize to a two dimensional grid of seats or, in this case, urinals.
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When is this discussion going to be central to an episode of Numb3rs?
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Who cares if they see your schlong?
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I would be rather interested in an optimal period cycle analysis, largely because I started my birth control (thereby period starts on day 23, ends on day 28) at a time where I was on my period for Halloween, Thanksgiving and Christmas. And how would different cycle lengths and different lengths of actual period factor into this? As any given woman’s cycle may be between 25 and 35 days, and her period may last 2 to 10 days.
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Finally. Math does something to help out us people with shy bladders.
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The female equivalent is totally choosing cardio equipment at the gym… just watch. No one will chose the treadmill or elliptical next to someone else if they can help it.
Of course, there’s always the added complication of who is on the rower behind said machines, or doing weights in front of them…
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maybe guys shouldn’t be so damned insecure about themselves and stop worrying about the guy next to them
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Dividers don’t actually solve the problem though.
If you’re in an empty restroom, using a urinal, if a guy walks in and takes the one next to you, you’re still thinking “why does he want to be so close?” regardless of a divider.
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“On the other hand, if there are seven urinals, they don’t fill up so efficiently:”
With all due respect, I believe that’s incorrect. Subconciously a respected gent will follow the protocol. If the far two slots are done, the smart urinator will take a spot one OVER from either end i.e. the third or third last spot. This minimises the chances of a fourth player coming and slotting right beside.
Aside from that, it’s right on.
AAB
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My buddy and I like “pissing off” guys at the urinals.
we find one guy in the middle of a row of 8 or more urinals and pick the urinals on his immediate right and left.
For that extra creepy feeling we will either both stare at the stranger or carry on a conversation as if he isn’t there.
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great blag-article! i considered writing a paper about the topic from a behavioral economics p.o.v…. 🙂
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Peter wish u never meet me. else i may aswell pretend that nobody’s there and try a 360 spin while at it 😉 u know cuz im all “alone”
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I wrote a small and dirty men’s room simulator a while back: http://wr3cktangle.wordpress.com/2008/03/12/bathroom-sim/
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Nice article, but I flaw, I choose the one farthest but one from the occupied urinal. Taking the furthest one suggests you are scared of the other guys wang. Unless there are only three in which I use the furthest one.
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For me, it has nothing to do with “size” being the awkwardness driver. (I’m not really into checking out the guys next to me.)
In a bathroom without urinal dividers, the closer I stand to someone, the more likely I’m going to get some “splash back” from the guy next to me. My own “splash back” is nasty enough, but standing next to the firehose draining 3 beers from his bladder and seeing little droplets of pee reflecting off the porcelain is just nauseating.
That and having to listing to the guy grunt and groan (ahhhh, oh yeahhhhh), is enough to warrant a minimum one urinal buffer.
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I always choose the “short” urinal….. that way if I lose my grip I’ve got a bit more time to catch myself before “it” lands in the bottom……
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I’d rather pee in my pants than use a trough. Ever go to a NASCAR race? Trust me – wait in line for the porta-potty. Either I use a stall or I walk out and look for another set of restrooms. First, I dont care for the “splatter”. Depending on the height or the length/dept of that stand up urinal, the splatter can be bad. On clothing – ON LEGS…….. f- that.
Stalls or no peeing. 🙂
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One important thing about urinal efficiency worth noting is that if you have two numbers with the same efficiency %, one will still allow a greater number of men to urinate. So if you were to design a bathroom with efficiency in mind, with room for 49 urinals, it’s going to be better than 3 in terms of the number of people you get through your bathroom, even though a lower proportion of urinals is being used. The downside, obviously, is it costs you a lot more to put in 46 more urinals.
Another good solution is the stall-wall-style dividers, which bring the efficiency to 100%, as they eliminate the need for buffer urinals, as they themselves are buffers.
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A lady quandry: You’ve had a tampon in your purse for a while, and its wrapper has opened. How unwrapped does it have to be before you declare it unusable?
Variables:
Extent of damage to ‘pon: from just a little tear to completely out and smelling of altoids or lipgloss.
Extent of damage to panties: are they already a lost cause? If so, am I wearing black jeans or a white linen skirt?
Arsenal: do I have anything else I could use? Is there a dispenser in the bathroom, and if so, does it only have those matress-sized maxi pads? Could a wad of toilet paper stem the tide?
Relationships: Are there other females around that I could ask? Is it my sister? My boss?
Timeline: How long until I’m home or somewhere else that such items are procureable? And, if I have to purchase, is it worth buying a whole box when I only need one?
Good luck!
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I take the first unoccupied odd-numbered urinal that’s at least two past an occupied one. If the first one is a shorty, I call it number zero, otherwise the first one is number one.
If, heaven forfend, I end up in an Awkward Situation, I make sure it’s more awkward for the other guy because he didn’t follow the protocol!
Sorry to pee on your heuristic… 🙂
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LOL, Nice
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You should call it the Irrefutable Choice of Urinal Protocol
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Melissa: that is why it is so much easier to be a male!
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This is all so enlightening, being female the choice is easy: pick a stall that is clean and if possible with no occupants on either side, but then the urgency often negates all that. You pee where and when you can, you guys are soo complicated :-]
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The number for ideal urinal arrangement corresponds to the sizes of groups which can be dominated by two people by successive committees. This can not be an accident!
What do I mean by dominated by two people? Take 17 for example. Imagine that the “council of the party” consists of 17 people. You suggest forming an “executive board” of 9 people, and convince everyone on that board that the board must present a unified front to the world, so that whatever the majority of the board votes for, the entire board votes for in the council. Within that group you create a presidium of 5 people, with the same arrangement of loyalty. Within that group you name the president, vice-president and secretary as the ‘executives’, and make the same deal within that group.
Now, if you are two people in the executive group sticking together, you control the council of 17, by successive votes. You will win the vote in the group of 3, the 3 will win in the group of 5, the five will win in the group of 9, the 9 will win in the group of 17.
Note that the groups do not have to be “higher ranking”, it may as well be political groupings within the groups “the democrats”, “the blue dog democrats”, “the blue dog democrats who eat children”, etc. As long as the principle of sticking together externally holds.
Urinals make powerful politics! Anyone wanna join the pee party?
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Nice research there urinal scientist…ur work cud ease of a lot of ‘wanna-pee’ guys, without the awkwardness ofcourse! LOL!
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Nice research there urinal scientist…ur work cud ease of a lot of ‘wanna-pee’ guys, without the awkwardness ofcourse! LOL!
piss off
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Great conversation starter, but blatantly plagiarized from Dave Barry’s Guide to Guys.
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International Choice Urinal Protocol? I C U- FFFFFFFFFFFUUUUUUUUUU-
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I’ve discovered that if the bathroom owner puts wooden panels at about waist height between each urinal, so that each urinal is a mini stall, they act as an Awkwardness dampener.
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I like to go pee.
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This was explained to me by my high school chemistry teacher, who quite elegantly tied it to Hund’s Rule.
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I will never be able to pee in bliss after this.
Someone’s got loads of time on their hands!!!!
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This protocol is ceratinly not international.
over here in pakistan we are forbidden to hold our thingy. so we have to go to urinals in pairs and co-operate. this protocol of maintaining distance wont allow us to hold our neighbours’.
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Great analysis
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need for further research on how the function will behave when the urinals are not aligned in a line… like modern bathroom we have on two rows facing each other or may be in L shape and how is the boundary conditions change with the Hand-wash basin being one side.
this is PHD subject …
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Peter, you’d do that stunt on me once and never again. You are lucky that you have not come across someone like me yet doing your childish game, you would at the least be missin some teeth by now…
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I like the posts that are sure this is a blatant rip-off of (insert author that you think is so original) Is it possible Maddox or Dave Barry aren’t the only ones who ever thought seriously about minimizing urinal awkwardness…..
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@Hallosaurs – Really? You’re so insecure you’d punch someone for peeing next to you? That’s one of the dumbest things I’ve heard today.
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This is hilarious..!!!!
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In the UK we often have a urinal that is one long metal trough, at which any number of guys may be expected to pee at one time. It may be worth investigating what is the acceptable buffer distance in order to avoid Awkwardness.
Do such urinals exist in other parts of the world?
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awesome post man!! and nice comments too..
@ Collin, not in India at least.
In urban parts of urban India – even with brick walls people religiously follow the urinal protocol
for rest of the places – People have enormous choices in terms of which tree to feed ammonia or which piece of land to irrigate. Here choices have always been need driven.
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Any research on optimal number of toilet paper squares to use?
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But what is more awkward? Pissing like a man who doesn’t care next to another dude in a packed set or waiting like a pussy to pee when there are empty urinals sitting there just begging for it?
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From a mathematical point of view, you’re right. (BTW, f:N->R+, so the graph can’t be continuous since you can’t have a non-integer number of urinals.)
But from the building owner’s point of view, this is a poor way to use urinals in the male bathroom: half of them or more are always unused due to the Protocol. Also, having very large bathrooms to accomodate 17 or 33 urinals is going to be an issue because any architect can tell you the rule of ladies’ bathrooms being at least twice the size of men’s. That’s going to be a lot of wasted space in the building.
So rather than installing a large, recommended number of urinals in order to avoid Awkwardness, I’d propose to limit Awkwardness by simply installing… vertical walls between urinals. By invalidating the assumption that there are no walls, the Protocol is not necessary either.
Also, the money saved from unused urinals can now be spent on “Please do not attempt to eat the mints in the urinals” signs. And a healthy stock of mints, because few people read bathroom ads.
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One might observe that sitting in chairs in public places follows the same patterns.
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