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 
Of course, they could just design urinals to be more separate as to eliminate the awkwardness.
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There’s a few variables that change the protocol. If there are sinks next to the urinals, if there are multiple walls with urinals such as in a corner, if there are dividers between them, and/or if the styling or shape of the urinal causes you to create more distance between you and the urinal.
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ad dividers and this is all pointless lol
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Or… you could man up and just piss wherever you want. If you feel awkward at a urinal, no matter the location of urinal neighbors, you’re just gay and afraid to get caught meat gazing.
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In ladies rooms:
To choose a stall I generally just check which one looks clean. If nobody else is there, I always pick the handicapped stall because it’s used less often, therefore rarely still has someone’s unflushed business or (god forbid, but it happens) backed up mess. Plus it sometimes has its own private sink, IDEAL for cleaning bloody messes in one’s pants. If people ARE there and able to see me choose, I peek in the other stalls, starting closest to the handicapped stall until I see one sufficiently clean. Finally, I avoid the stalls closest to the door, because they are so heavily used that they usually look and smell worst.
To avoid awkwardness when leaving, I listen to the doors and the sink. Ideally, nobody will see me leave my stall so whatever noises I made are anonymous. If I succeed in that, I listen for flushes so I can hurry up and finish washing my hands so I can get out quick and afford the other woman just such anonymity.
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Okay, If your the First Guy you go to one of the End Urinals, Second Guy goes to the Other End Urinal. Now this isn’t Rocket Science But… The next Dude in should go to any Urinal that maintains at minimun one Urinal spacing, more is Okay, but not less. If more Guys Show up Always maintain the one Urinal Space ( When Possiable ) But when you have to go, you have to go. Just go in and concentrate on your own thing, get done and yield your Urinal to the next Over Full Bladder. And if your just in there for the Thrill of Meat Gazing, you should have the Crap Beat out of you. Have a Great Day!
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I never care about that protocol nonsense. I just try to find the urinal that has the smallest puddle of pee on the floor under it (or no puddle). It’s rather disgusting: some of those puddles are so big that I wonder why they even put toilets in the restroom at all.
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I would just like to say that the guy who posted “JUST DON’T CROSS STREAMS DUH” is a genius! I have never thought of that before. This is an astounding way to view to situation. I have more to say and would like the personal urine sample of this guy, but I am too excited. I must review my theories and revise them in light of this new perspective.
Thank you ‘thisguy’. You don’t realize how much you’ve helped the scientific community.
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This apparent inefficiency calls for a New International Choice of Urinal Protocol 2.
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This is why I firmly believe all urinals should have dividers (to head height preferably, shoulder height minimally) between them.
And those long trough urinals you see in some places are Awkward Central… especially when somebody with no shame or too much alcohol decides to try to “squeeze in.”
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Please expand analysis to include the ‘trough’ – a urinal of finite length but no stalls. Basically one long urinal, usually seen in football (soccer) stadia.
Possibly the most efficient urinal as one can always slot into the mid-space between two existing participants. The limit then becomes a function of Awarkdness (which may in turn be a function of beer consumed at game) and shoulder width.
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Obviously, it’s better to set up the urinals with a small divider wall in between each urinal. That way, no urinal is truly awkward. That’s more peeing men per wall meter than if only every other urinal is used!
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Just as a little note. The worst case formula is more complex than needed.
n = (3/2) * 2^k + 1
Can be rewritten:
n = (3/2) * 2 * 2^(k-1) + 1
n = 3 * 2^(k-1) + 1
because k just represents any integer, k-1 could actually be written just as k:
n = 3*2^k + 1
But anyway. At least I used to add some randomness to the choise of urinal. Lets say I follow that pattern in general, but in 50 percent of the cases I choose one urinal at the side, as long as it isn’t right beside someone else. So the question is, let’s say noone leaves his urinal, how often does n urinals pack to 50%?
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The only building on campus where I’ve seen urinal dividers is the engineering building.
Coincidence?
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There is another factor that i feel has not been considered here, when arriving at a urinal and faced with a potential awkward/non-awkward position, there’s the ‘prediction of departure factor – i.e. that it might be better to take a moments awkwardness, and stand next to someone who looks like they’re about to leave, rather than risk a potential ‘closeness’ on both sides if one goes to the vacant position that currently has space on both sides.
Essentially this formula follows from the starting position of T=0, however whenever we enter the toilet, our time might be zero, but the status of residency is not.
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A more useful and practical calculation would be the optimum ratio of urinals to stalls to provide the least amount of waiting time.
In a lot of places there can be a large number of urinals and only one stall which makes little sense as the time needed in a stall is much larger than at a urinal.
Considering stalls are multi purpose, I think sacrificing even three urinals for a stall would be a fair trade.
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Its funny how readers fail to see that this entry is a direct rip-off of Maddox’s book “The Alphabet of Manliness”.
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http://www.fukgames.com/game/4385/the-urinal-game
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OK, apologies if someone else has already posted this, but I didn’t notice. Another factor is the dominance factor of the arrivals. This predicts that those with the greatest need for dominance won’t go the the end, but will choose the most dominant position – this would usually be the middle urinal of a set of empty ones. Conversely, the meek would always hide away as far as possible. Strangely, with already largely occupied urinals, the two behaviours lead to the same result, but with large gaps, especially at the ends, they will be different.
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check out this video:
http://www.youtube.com/watch?v=KnOaMC8KHA4
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also, this theorem totally ignores the fact that the subjects choosing the urinal are humans. if there were 7, and the ends are taken, I would never take the middle. not sure that I ever consciously considered why, but now I know, because obviously this increases the chance the next bloke will stand next to me, whereas w/ 7 if you move down a bit, then your chances fare much better. therefor, granting some sentience to the subjects, 7 urinals is just as good as 8.
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See I have a massive shlong so I dont care who pees next to me.
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May be this video is useful:
http://www.youtube.com/watch?v=KnOaMC8KHA4
and what about such installations ?
http://twitpic.com/puks
this would mean n != 1, n= ?
^^
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Along with the alcohol and dominance factors already mentioned, there’s the penis length factor to be considered!
The shorter, the further away from others…
Some long sized men will not consider awkward to stand on any position, even if it’s close to a another guy. I know I don’t. ;P
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Suck it up, ladies.
really, it’s not favourable to pee next to another bloke, but MEN don’t care.
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In the military, I’ve actually had to share urinals. When 70 guys need a pee break and only have 10 minutes and 3 urinals..
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I have come up with the two perfect solutions to this problem:
1) Be born with a large member. That way you don’t mind where you go and you secretly want to intimidate all the other less endowed guys around (It’s a caveman thing).
2) Get drunk. You won’t care what is going on by that point.
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What happens if the urinals aren’t disposed on a line, but form -just an example- a circle? or a Moebius strip? Or what about an infinite number of urinals (well, actually, that’s more like the Hilbert Hotel, and the solution is easy and not very awkard)?
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Please note that this protocol doesn’t apply to gay people or gay bars.
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Sequence A095236 in the OEIS considers another aspect of this problem.
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The only female related issue I can think of… and will most likely cause queeziness for the men out there. Is the frequency at which to change your… ahem… tampon. Varies by day, product choice, and your timing may be bumped up or down based on meetings, bathroom availability and cleanliness, etc.
Or – getting pregnant. That would probably be of more interest to the men – the mathematics behind knowing when your girlfriend will NOT get pregnant.
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I’m surprised nobody has mentioned potential for a ‘man-in-the-middle’ attack.
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i think this is pretty revolutionary…and i guess I shall strive to minimize the awkwardness…and ofcourse at time poke fun by making things awkward…
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I guess as a female, I never pay attention at how guys would choose a urinal. One thing for sure, my husband never use a urinal unless in desperate times. He always go for a stall even though just for a pee.
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A surprising analysis to be sure. Interesting the coefficient of awkwardness is dwarfed by the coefficient of “scared of bears”. This is evidenced by boy scouts on camping trips. They will, invariably, all choose trees as urinals where they are positioned to hide their privates, but close enough to ensure that even the slightest whisper of “did you hear that noise in the bushes?” can be heard by all simultaneous ‘relievers.”
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Very sneaky, but we’re still not going to tell you what we do at sleepovers.
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I agree with “Stewie” – real men don’t care about this bull … what kind of panty-waist little boys fret about peeing next to another man?
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I’m surprised whenever people think the urinal protocol is about penis size. It’s a matter of tact regardless of size. It’s the awkwardness factor, not the shame factor. Besides, every man is looking at a space 10 centimetres up and 5 centimetres to the left of eyeheight while peeing. So penis comparisons should not be possible, except by tragic accident.
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Isn’t this basically the macrocosmic equivalent of the spread of valence electrons, where awkwardness equates to electron orbital configuration stability?
I have had a chem class where the tendency of valence electron placements was described using a similar analogy.
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I usually just ignore it. My choice of urinal protocol is the closest I know works and isn’t disgusting. Much easier to calculate.
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What a bizarre coincidence – I wrote about the exact same topic the same morning. Check out moralhazard.wordpress.com/2009/09/02/mens-room-mathematics. Many of our conclusions are the same, but I defined the function recursively rather than in closed form. I also noticed a very unusual pattern in the sequence of “effective urinals.”
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Can you extend that to the ditches usually misused for this purpose at your average open air music festival?
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Very good explanation of the Buffer Urinal Theory: http://www.bufferurinal.com/2008/10/why-buffer-urinal.html
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Why not organize bathrooms such that urinals have a “urinal-distance” space between them? Or, organize fixtures in a urinal-stall-urinal pattern? Or, embrace the awkwardness.
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This may have just replaced “Little Bobby Tables” as my favorite…
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I remember going to the men’s room in Toronto’s Union train station in Canada. The urinals there were basically toilet bowls without the seat. And those urinals had no wall to seperate or “hide” its users from each other. So imagine pissing in this toilet bowl that is sticking out, wayyy out without any privacy.
That was the top of all awkwardness for me and I just used the stall instead. I was not gonna piss in a bowl arranged for everyone to see.
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