# Urinal protocol vulnerability

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.

## 1,135 thoughts on “Urinal protocol vulnerability”

1. Very interesting.

In Australia in smaller pubs you’ll often find a trough that could hold 3 people max, and about 2 toilet stalls.

Which sort of brings up another question, the etiquette when pissing in a real toilet stall rather than the trough.

Almost every single pub you ever go into has toilets that you wouldn’t want to shit in, because there is that much piss on the seat and especially on the floor that it’s just almost impossible to do a shit. The seat you can clean off with toilet paper but the puddle on the floor will see your dropped pants soaking up a cocktail of previous patrons urine. Not ideal at all.

But sometimes you’re at the pub and you really need to drop the kids off at the pool. It’s quite a conundrum. You could put more toilet stalls in but I feel that people are still going to ‘evaluate’ which of the stalls is cleaner looking, has a functional door lock, etc, and hence the piss will find a way to reduce the poo-functionality regardless of what you do.

One way to fix this, though perhaps extreme, may be to make all urinals have irrefutable privacy such that even the most shy person can piss into a urinal with a door closed behind them, and walls beside them. That would remove the only ‘advantage’ the bathroom stalls have over the trough, and thus should see the actual toilets only used for number 2’s.

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3. curiously enough both cases follow the sequence:
x=2x-1
for numbers greater than 2 (best packing) and 3 (worst packing)
take that fibonacci! did my sequence piss you off? ;D

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4. But… that’s wrong! The protocol doesn’t include end urinals. It’s just simply that you leave one urinal between you and the next guy. In a group of seven, you wouldn’t do that unless the toilet was completely dead. If there were queues, 4 folk would fit in there just fine.

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5. This study should be developed. What about urinal *lines* in room corner. Optimization of the furniture configuration: washbowl, hand-dryer etc… I guess it would be messy putting the washbowl between two urinals.

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6. Great website…and cool article man…thanks for the great post…keep on posting such article…

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7. Considerably, the article is in reality the sweetest on this notable topic. I am glad that I found your blog in Google, I will back here often.

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8. So… let me get this straight. Say there are seven urinals, and they’re packed (efficiently, I assume.) Let’s also say there are no toilets. Would guy #5 use a sink or wait before using an inbetween urinal? How much of a faux-pas would it be to create Awkwardness?

Us women folk don’t know these things. We do know how difficult it is to ninja-poop though (meaning either nobody knows there’s pooping going on or nobody knows it’s you).

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9. Article needs an addendum for the single trough urinals that are used communally. There has to be a different protocol for those.

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10. @Dave:
Its the continuous case of the same problem. Instead of having a Boolean definition of awkwardness, awkwardness could be a function of distance between people. Each person could have an individual maximum of awkwardness (i.e. some people can tolerate a greater amount of awkwardness than others), and so packing efficiency would depend on many other factors as well.
Perhaps a numerical simulation is in order?

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11. … or you just have a trough which is divided and conquered as per simple rules until the space is too awkward. Then things get rainbow-coloured.

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12. Mhm, I’ve never heard of this protocol, only the simple rule that you don’t take a spot right beside someone. Which makes me wonder why I see so many bathrooms having only two, as only one can ever be used. (I guess you get redundancy…)

Of course, the real solution is to put dividing walls between each one. This eliminates the problem entirely and provides a convenient place to communicate phone numbers and call people fags.

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13. …and whatever you do, do NOT drop your pants all the way to the floor like a 2-year-old would. VERY awkward.

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15. Well… urinals are commonly found close to places where people get drunk. It’s widely known that drunk people do not adhere to any protocol in the urine area.

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16. Thx for that well job.

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17. An Airport Corollary should be developed that takes into consideration a large number of urinals with little or no activity and an over whelming influx of customers when a flight arrives…… urgency(x) = awkwardness

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4 is one of the ideal configurations, and so is every even number higher than 4. Too bad a circle of urinals would take up much more space than a lign of the same number, but if there’s enough, you could have the stalls and wash bowls in the center. It would be like a Panth-pee-on!

I think I’m getting carried away here…

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20. Women use the theory of least usability. If you are a non-toilet liner you want to assume the most un-used toilet. Stall #1 is generally the most common used for frequent uriners, so because of their urgency, may not have been as neat about their peeing procedure.

Then one must guess as to which other stall may have been her counterparts choice stall, a simple observation might have been remainder of toilet paper in the stall, however one must interject the possibility that the toilet has recently been cleaned and if the stall was out of paper, would now have a new roll. Since rolls aren’t dated, it’s not easy to assume based on the amount of remaining toilet paper.

The handicap stall is used almost as much as the first stall and increasing the possibility of used more the we’re discussing bathrooms in a gym or fitness area requiring people to change.

Deductive reasoning; using toilet seat covers saves you from thinking. They can also double as oil removers for your face, just use one clean one for your bum and one clean one for your face.

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21. Wonder what would happen if you place urinals seperately bu in a circle…

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22. I’m thinking that this is a post from someone who may have too much time on their, er, hands. So to speak.

I say avoid the problem altogether and use a stall.

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23. While Urinal Etiquette is very real, and this is a good write up for it, it should be noted that some urinals have partial walls between them, since the view is blocked and it’s more stall-like, the normal etiquette can be ignored, especially if there are fewer urinals and the only empty one not next to someone is the lowered urinal for kids. It’s only common sense that when you have a stall-like situation that you can be next to an occupied urinal, granted if there are many open it’s still a good idea to use proper etiquette

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24. Women use the toilets that don’t have pee/hair/toilet water/unidentified substances on them. I personally love the roomy handicap stalls.
Also, women love going to the bathroom in large groups. We talk about cramping, ask each other for spare pads/tampons, complain about getting fat then reassure each other, and probably talk about whoever else is with us but did not join us to the restroom. If a girl goes to the restroom and is in the presence of other girlfriends, and does NOT ask one of them to come with her, it’s very likely she is taking a poo.

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25. this is a picture from Hacked irl (an icanhazcheezburger site), thought it was worth mentioning…

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27. most if not all awkwardness would be removed if floor-to-ceiling dividers were installed between all urinals.

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28. You rock!

I taught about that so many times… trying to predict witch urinary would be the one least used and obviously the cleanest :-)

I newer taught someone would be crazy enough to put that theory on the paper!

You also have to take toilet entrance position into consideration :-)

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29. Better yet, put dividing walls between the urinals, and a door behind them. And once you’ve done that, you might as well reconfigure the urinal to accept solid waste as well. Oh, wait.

Seriously, what moron came up with the urinal?

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30. All this does not address the situation of gang urinals — the troughs found in some railway stations and even pulic schools in the past. You can stand any place along those and still catch an eyeful if that’s your thing. Desperately modest people an always wait for a stall, but in a crowded, busy public toilet, not many people will be intrusive of someone’s privacy.

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31. Two urinals, one corner. Urinals placed back to back so while urinating, people are facing each other (possibly making eye contact). Double-decker bathrooms. Urinals on a conveyor belt. A configuration in which you remove all non-awkward toilets. A wax sculpture of someone peeing (perhaps with their head against the wall and eyes closed).

Seriously, I’m hoping someone who is currently designing a bathroom reads this and is inspired.

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32. Easy, make higher divisions between urinals, so U don’t have to see the other persons “face”. Let’s say, about 8 Ft is enough…

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33. Forget seeing each others face. its actually embarrassing and awkward to pee next to a stranger.

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34. some one said shouldn’t see each others face while peeing, Forget seeing each others face. its actually peeing next to a stranger itself is embarrassing and awkward .

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35. Someone said shouldn’t see each others face while peeing, Forget seeing each others face. its actually peeing next to a stranger itself is embarrassing and awkward .

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36. What about urinals with those stall divider things. does protocol apply then?

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37. The solution is simple. The Pattern goes: Urinal, Stall, Urinal, Stall, Urinal, Stall.

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38. Some of you guys have NO idea about protocol.
NO eye contact, EVER. And no tapping of toes, or dropping of combs or other objects. If you drop something while peeing, you’re trying too hard, and are highly suspect of having motives other than purging your bladder.
Assuming that no eye contact is made, then, everybody in the room is on the same page, and it’s OK to leave without washing your hands, assuming that your mother taught you not to pee on your fingers, because guys that are so prissy that they HAVE TO wash their hands are spending too much time in the Men’s Room, and are generating Awkwardness. Urinals outnumber sinks by a wide margin, typically 2-3: 1. Also, it’s not manly to poo in a public restroom unless you are in imminent danger of crapping your pants, on a crucial first date with that hot little number that just got hired in Accounting. If you “have to” take a dump, McDonald’s is generally considered the place least-likely to have some horrific insect infest your rump while you’re there.

These are all matters that fathers should cover with their sons, before the birds& the bees.

HTH and Regards

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