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 
or there could just be walls in between them, like in some places
LikeLike
I have been trying to explain this problem to women for some time. This article will help so much thank you.
Robin.
LikeLike
This just occurred to me: there is another application for the best case equation and that is hanging your drapes (curtains). Start by clipping the leftmost and rightmost edges on and proceed to recursively split and clip the remaining area(s). For this to be effective, you need 5, 9, 17, 33… curtain clips.
It’s also interesting how this is essentially just a perfect binary tree with two additional clips.
LikeLike
It occurred to me last night after the movie that there’s another variable that creates issues: Child-height Urinals
The bathroom at the local theater has an optimum number of urinals, but one is child-height, and average-or-taller people tend to shy away from those, for good reason, too: They’re not comfortable if you aren’t child-height.
The Urinal arrangement was:
C, A, A, A, A (C being Child-height, A being Average-height)
When I went in, someone was at A4, and my friend ended up taking A1, fitting within normal protocol. For lack of an optimal urinal, I ended up at A3, theorizing that it would be better to be awkward with someone you don’t know than someone you do.
How do Child-Height urinals factor into the protocol? Should we take the dive and use a Child-Height urinal if it will help the people who come into the bathroom after us avoid awkwardness, or is it acceptable to just treat the Child-Height urinals as non-urinals and settle for the awkwardness of a 4-urinal bathroom as opposed to the idealness of a 5-urinal bathroom?
LikeLike
Okay, solution for the 8-urinals problem:
We have as shown above, but number five uses BOTH urinals 2 and 3. Therefore, no guy stands _next_ to each other, but there is the problem of correct aiming.
Therefore, the best solution is the “long trough” urinal system.
LikeLike
All right, i am all for a buffer zone between me and someone next to me who may “shake” a few too many times but if i need to take a pee and i have to stand next to some guy to do it…. well i am gonna pee, and not go hide in a stall to pee.
seriously … are we men or boys, why do i go into public (or even office bathrooms) only to find some stooge hasn’t figured out how to spray his pepper pee soup all over the toilet see.
If you are goning to go into a stall… don’t miss, i mean you have had that thing for a long time. Probably peed atleast 3 times a day for years … thats 1000 times a year… and you still can’t aim.
sorry for the rant, but really.=, point and shoot, not point and spray.
LikeLike
I raised this issue a few years ago here, though not so thoroughly:
http://www.gaire.com/e/f/default.asp?page=view.asp&parent=232082&nav=1&bot=
LikeLike
My friends and I are intrigued. How does one (a mathematician or an algorithm wizard) come up with a closed-form expression for the number of urinals that can be utilized without awkwardness? It’s a pretty involved expression, by my standards.
LikeLike
when man needs to do it he tries to find proper place
if he cant
he wil do anywherre
LikeLike
I think child-height urinals are the biggest problem facing urinating men today. My office has only three urinals, one of which is child-height. Obviously, I have never seen a child in the bathroom because it is a restricted-access office building. Worst of all, bathroom planners always put the child urinal on the end, reducing an efficient 3-urinal setup to the worst of all–the 2 urinal.
Here is where it gets good: We have barriers between the urinals, but when the end urinal has been taken, nearly all men choose the awkward child-height urinal instead of the man-sized urinal next to the occupied one. I, being 6’4″ laugh at their fear of social pressure and choose to pee at a comfortable height next to the occupied slot.
LikeLike
Are you trying to imply that the protocol calls for guys to wait for a buffered urinal? I mean of course there is optimal urinal choice, but when I gotta piss I gotta piss and dont care if I have to go between two dudes…Most places will have something to read or even tv screens above the urinal, so if someone is taking a peak instead of reading/watching tv you should be pretty much unaware.
As for the child urinal, I agree they are the last choice but big deal, a little splash back never hut anyone.
Personally I would rather pee outside weather it be on a tree or behind a dumpster. I can even start and finish a pee while walking with no problems.
LikeLike
Seriously, Madox, alphabet of manliness, this is already covered. Not in such mathematical complexity of course.
How do you respond to this as a dynamic system? Each section changes appeal depending on timing. say we again use the 6 slot system. Should 1, 3, and 5 are filled. The next slot chosen should be 2, depending on the average time and the P-shy factor. This is because of the expected timing of slot one becoming non-awkward in the shortest time. The Pee-shy factor will linearly increase time spent on a personal basis (as a starting transformation) and exponentially increase with higher populations, to a certain number. At this point one of three actions is taken; take a stall, just pee with your eyes shut, or leave admitting defeat.
At this point 9 looks to be the optimal size for larger restrooms. slots 1,4,7 are first taken. Then 2, 5, and 8. By this point there are no non-awkward spots left, but 2, 5, and 8 were better as they were less awkward. The final spots should not be filled unless there are serious line issues. HOWEVER, timing of 6 slots being taken allows more than ample time for 1 or 4 to open up, depending on temporal expansion for pee-shy and alcohol considerations. After this the possibilities are near endless, but the optional cyclical nature of it allows for minimal personal analysis before choosing. You wouldn’t want to look like you were watching or anything.
LikeLike
Also this is what C3-PO is programmed to deal with. Protocol in human interaction. I hope we never offend some alien race who think it’s the best place to chat.
LikeLike
Ahh piss on it (pun intended)! If there is a free urinal and I have to go, I’m taking it. Of course I will swivel away slightly from an adjacent occupied urinal.
LikeLike
As long as there are separators between urinals, than there really is no problem… Restrooms without separators make me cringe!
LikeLike
I find that if all else fails in this case, use a cubicle.
LikeLike
It’s funny that I stumbled this right now. Last night I was at a party discussing this same issue with someone.
LikeLike
It’s funny that I stumbled this today, because last night I was discussing this same issue with some one at a party!
LikeLike
using a stall or cubicle only works is the seat is left up. Who among you wants to lift a nasty old toilet seat. (Its different if you need to do #2, after all you’re going to sit on the bloom’n thing.
LikeLike
The solution is easy, just follow the protocol but take only the urinals with odd (or even) number 🙂
LikeLike
This problem becomes even more complex when you consider certain exceptions.
Two guys discussing the ‘game’ will almost always do it next to each other. (As long as they are talking the will be looking face to face, no lone looks down.)
Also father swith young sons will also occupy adjacent urinals.
LikeLike
Good video to supplement the post:
LikeLike
LikeLike
In the subways (New York City at least) it’s the same protocol. Someone sits on one end of the seats, next person on the ohter end, and others in the middle.
LikeLike
am i the only one who uses the sink?
LikeLike
i’m sorry, but is it really that ‘akward’ to pee next to a man? if you gotta pee just pee. now, if i were to magically convert into a male i would most definately pee right next to the nearest man. calculate that theory now boy.
LikeLike
No ian, I use the sink too…
LikeLike
This is funny because i have just read it, and now i have to go to the restroom.. maybe i could test it out
LikeLike
Urinal hack!
LikeLike
Can you also address issues of wear and tear? i e if you have five urinals, you use only 1,3 and 5, but 2 and 3 don’t get used.
Consequently that can be seen as a waste of equipment. Do some numbers have better distribution of wear and tear on the equipment than others?
LikeLike
Addition to drunk peeing rule:
If drunk enough some people won’t even bother using the urinals. They’ll stand up on the bar, take out their wang and pee every which way, offten saying unkind things.
LikeLike
I think it’s common knowledge that in most restrooms Urinals #2 and 4 are non-functional display models installed to set the spacing for the real ones.
LikeLike
Actually, it’s getting worse. We 50+ animals don’t really have a problem fluting next to another, but the younger males . . . I regularly see them take a cubicle if there’s even *one* guy at the troughs. Probably don’t play enough sport.
Then there’s the issue of female urinals for stand-up peeing to alleviate the long lines in female cubicle toilets. I’d like to see an algorithm for that!
LikeLike
One possible (though complicated) expansion to the algorithm could include the placement and use of urinals at different heights. I for one find it awkward (regardless of spacing) to use any urinal where the top of the urinal is less than 36in./91cm from the ground, and if this is the only non-awkward urinal available, I opt for the stalls instead. The numeric figure I provided is based on my own body dimensions, so such an addition to the algorithm would require a statistical analysis of all the men who are likely to use public urinals, and when the top height of one of the optimally-placed urinals is shorter than the mean crotch height, it can be removed from the set of available urinals according to the proportional preference/drunkenness of the next person in line.
And if my aspiration of becoming an architect ever comes true, I plan to place the shorter urinals in the non-optimal spots. Doing so might even force efficient packing …
LikeLike
Putting small dividing walls between each space is the best solution. You get full use of each urinal, and it takes only slightly more space to achieve the effect. A friend refers to this tactic/exception as the “Jeopardy! clause.”
LikeLike