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.
Wow… In my experiences in the men’s room, if the urinal count is odd and over 7, it’s safe for (urinals / 2 – .5) people.
Despite the lack of realism in your “rules,” it’s an interesting idea, and it generates an interesting series.
I think your formula is wrong though.
I wrote a program to simulate your scenario, and I get the following, where the first number is number of toilets, second the max capacity given the rules, and third the efficiency. These numbers differ from your formula and your graph; nonetheless, I’m confident they are correct.
1,1,1
2,1,0.5
3,2,0.66666667
4,2,0.5
5,3,0.6
6,3,0.5
7,3,0.42857143
8,4,0.5
9,5,0.55555556
10,5,0.5
11,5,0.45454545
12,5,0.41666667
13,5,0.38461538
14,6,0.42857143
15,7,0.46666667
16,8,0.5
17,9,0.52941176
18,9,0.5
19,9,0.47368421
20,9,0.45
21,9,0.42857143
22,9,0.40909091
23,9,0.39130435
24,9,0.375
25,9,0.36
26,10,0.38461538
27,11,0.40740741
28,12,0.42857143
29,13,0.44827586
30,14,0.46666667
31,15,0.48387097
32,16,0.5
33,17,0.51515152
34,17,0.5
35,17,0.48571429
36,17,0.47222222
37,17,0.45945946
38,17,0.44736842
39,17,0.43589744
40,17,0.425
41,17,0.41463415
42,17,0.4047619
43,17,0.39534884
44,17,0.38636364
45,17,0.37777778
46,17,0.36956522
47,17,0.36170213
48,17,0.35416667
49,17,0.34693878
50,18,0.36
oh nevermind perhaps the results are the same
The current model neglects time entirely. The original seems to assume an infinite duration of pee, and only one pisser to be added. The refined method of moving off-center to create two perfect gaps does account for additional pissers, but still figures they all are wearing camelbaks.
In a real situation, is it not acceptable to intentionally choose a urinal which is initially awkward if neighboring urinals will soon become vacant?
A better model would account for this, but it must incorporate acceptable proportion of awkward time, average pee duration, and average pee frequency.
It should probably also include how bad you have to pee, because that is positively related to your own pee duration. Everyone else’s pee duration is most likely unknown, but perhaps their time of arrival could be estimated (assuming they obey the advanced model as well).
One important question is whether it is truly the proportion of awkward time that matters, or a maximum duration of awkwardness.
Every since I’ve learned that I have an xxlarge unit, I like to take the center urinal to show off my package.
Don’t get me wrong, I’m not gay but I do like to show the other guys up.
It’s like having a sixpack abs but better
have you considered a workaround for the architects in order to maximize comfortable spacing withotu sacrificing empty urinals?
there must be a distance which males are comfortable using a urinal within. is it the space of a urinal? is it less? otherwise that’s a lot of porcelain and water systems being neglected for the sake of homophobia
or perhaps this could all be avoided with walls between urinals, which extend out into the room rather than just enough, so that males aren’t forced to look at each other at all
Awesome. I’m totally going to use this as a reference for an architectural design assignment the next time I have one at university.
Jack – I’m not sure this is based on homophobia, so much as a general Awkwardness arising from peeing in excessively close proximity.
The few gay bars I’ve been to had either 1 or 3 urinals. The few times I peed at the one with 3 urinals, ICUP seemed to be followed at least generally – the end urinals were always chosen first, the centre one only last.
This might all be avoided by using the single-long-trough urinal design sometimes seen in stadiums and behind rural Mexican bars (seriously, I first thought there was just a bunch of guys peeing on the wall – turned out there was outdoor plumbing specially installed for the purpose).
Or does this just create infinite Awkwardness, by forcing all peers to share one urinal?