Thursday, 7 April 2011

The Bottom Line: which cubicle?

One might think that a couple of months in Australia would be relaxing and refreshing, with freedom from the worries of everyday life. Unfortunately, it's one difficult decision after another.

Playing The Odds: In Search Of A Royal Flush
Yesterday, I was faced with a serious problem. I badly needed to take a dump. At home, that's not too much of a problem: realistically there is only one toilet that can be considered adequate, so there is no decision to be taken. Also, there is no danger of contamination: there are only two of us who live there, and I have reluctantly learnt to embrace Helen's bum as if it were my own. But I was at the University of Melbourne, and the situation called for some careful thinking: five communal toilet cubicles, and I knew nothing of their history. Where should I park?

As you enter the little boys' room, you are met by five cubicles on your left, with sinks on the right. The cubicles are not all equidistant from the entrance; and this is what gives the problem its crunchy texture. How do you find the least-used toilet?

Let me present you with my own reasoning, and then give you a chance to decide for yourself.

Pure Gamble? Let's Shoot Some Craps!

Obviously Cubicle 1 is out. It will receive far too much traffic from unthinking visitors who rush in where angels fear to sit. Pass by on the other side.

Doing one's business in Number 2 would have a poetic ring to it, but would also be misjudged. Plenty of customers will no doubt sensibly rule out the first toilet, but take the analysis no further, and dive head first into the second one. Besides, whenever the first cubicle is occupied, thoughtless types are going to be drawn to the second.

Now it gets harder. I suspect that the furthest cubicle would attract the naturally reclusive and socially withdrawn; and there are plenty of them in a typical computing department. So it, too, probably gets more than its fair share of attention. Number 5 would be a bum steer.

That leaves us with Cubicle 3 and Cubicle 4. What to do? It is tricky to make a strong case for one over the other. For a while, there was a serious danger that I would stand fixed for ever, equidistant between the two cubicles, like Buridan's famous ass. But at least the worst his ass had to look forward to, in the short term at any rate, was getting a bit peckish: my own ass, analogically speaking, was likely to force the issue if it didn't get some outside direction soonest. I had to make a choice.

Why would someone end up in Number 3? There are two plausible reasons:
  1. Someone is occupying Number 1 (highly likely), and our new chap wants to place a respectable distance between himself and the current occupant of Number 1. My guess is that in this case he would be likely to go for Number 3 (sufficient distance) or Number 5 (as much distance as possible).
  2. When all cubicles are empty, going in Number 3 preserves the symmetry. It may well appeal to a certain mathematical way of thinking that could be prevalent among computing academics.
Neither of those is a watertight argument, but on the other hand, I couldn't see any reason at all why someone would end up in Cubicle 4.

So, flying, to some extent, by the seat of my pants, I made friends with Number 4.

But I couldn't sleep last night.

The nagging thought: did I get the wrong one?

Have I embarrassed myself with foolish reasoning?

What would you have done?


  1. You've made the right decision for your own value system, but you have, sadly, broken toilet etiquette. See for a rough guide.

  2. did any of them have baby-wipes?

  3. I recently came across this problem on the cross channel ferry, however I had 8 cubicles to choose from! While your analysis was good for a 5 cubicle room, I feel it left the reader on their own for larger, or indeed smaller values of cubicle number: I have thus taken it upon myself to construct a quick frequency distribution function...
    F(n)=0.25*( [2/n] + [1/φ] + [0.5^(N-n)] )
    where φ=1+2(|N(1/2)-n|)
    N represents the total number of toilets available, and n is the particular one you are interested in finding out about...
    The 2/n term is representative of the unthinking visitors and others who will take the next occupied seat-it will have a maximum value of 2 rather than 1 as in the other 2 terms since I believe it would be a good approximation to assume that half of the population don't care which toilet they use, so will just go for the nearest one
    1/φ represents another quarter of the people-the symmetric people, however they are unlikely to sit near the central cubicle if it is taken, hence the way phi is defined such that the adjacent cubicles are unlikely choices, if you're really fussed about symmetry you can probably wait a couple of minutes...
    The final term is then for the final quarter-reclusive types, but these people I would assume would not take the N-1 toilet if N is occupied maybe choosing to wait, or come back later, hence this follows exponential decay to account for the unlikely chances of the N-1, N-2 etc toilets being chosen by them.
    If I've done my maths right that should give you a number between 0 and 1 for your chosen toilet-1 high 0 low. Note-it isn't a probability, just a scalar value for toilet usage. It makes sense to me on my bit of paper I just scribbled it down on anyway... I think.

  4. Your first choice was ignored: chance or logic.

    Given a set of indistinguishable rational actors, their logic will be the same and lead to the same conclusion. Thus if a rational actor chose to follow logic, as you did here, they would all end up in the same stall. This is true regardless of the scenario and the trail of logic. Given that you are in a computing department, the indistinguishable rational actor hypothesis is a good approximation.

    So the answer is to select a cubicle at random; since all rational actors will also come to this conclusion, the traffic will be even to all cubicles, and it ceases to matter which you choose.

    If you really want, you can exclude trap 1 because of all the visitors, but I think you're just giving in to fear; no-one goes into the computer science department accidentally. Even in Australia. The other users are all just like you.