Uninformed Comment

Pick a number, any number

Posted in Mathematics, OpenSim, Psychology and perception by uninformedcomment on June 8, 2009

“Pick a number, any number between 0 and 65,536.  Now pick another in the same range.”

If you asked that question of hundreds of people, and plotted their pairs of numbers on an XY graph, would you expect a roughly random distribution?  That question has been accidentally asked in the real world, and the answer may surprise you.  I’ll explain.

I wouldn’t be the first to remark how bad human beings are at dealing with large numbers.  Anyone who’s even dabbled in evolution, astronomy, geology or other fields dealing with huge numbers will be familiar with the kind of blindness that comes from trying to conceive of those numbers mentally.

Thinking

I was reminded of this a while ago in the context of the virtual-world software OpenSim, in a discussion in the forums of the OSGrid network.  OSGrid is one of a number of grids of OpenSim regions, each region being a square of land forming part of the overall world, or grid.  Anyone can install OpenSim and attach their region(s) to the grid, picking their own free spot by choosing its X and Y coordinates.   The coordinates they pick will dictate their regions’ placement on the grid, thus the neighbours they have, if any.

So, when someone creates a region, and it comes to deciding where to put it, they have to take into account where other regions are.  For one thing, they can’t put a region in a place that’s already used.  For another, they may or may not want the region to be adjacent to others.  Being adjacent has both its advantages and its drawbacks: it makes it easy for people to walk across onto your region (handy if you want to encourage visitors), it makes your region look less isolated, and so on.  On the other hand, it affects the performance of your region adversely.

OSGrid map sample

L and I currently run between 7 and 1 1 regions on OSGrid.  I host them here on a spare PC, and L connects to my PC (and to OSGrid at large) through the internet.  When I first set those up, the choice of neighbours or not was clearly answered – we weren’t particularly bothered about attracting visitors, and we wanted to use our limited resources to the maximum on other things.  So we decided to place our little cluster of regions far apart from others, and picked numbers “at random” in the range allowed – 0 to 65,536.  There are currently 1,822 regions, including our 11; we are almost literally a drop in the ocean.

The reason I tried to pick numbers randomly, rather than by looking at a map of the existing regions for spaces was the the mapping tool provided is pretty crude.  You can’t zoom out far enough to see more than 80 regions at one time: a very tiny window on a pretty massive area [UPDATE: this has since been corrected by new mapping software – UC].  In all, there are 4,294,967,296 possible regions.

I thought about how region coordinates people picked always seemed to be low, even when it was plain they’d been looking for isolation, and furthermore that they seemed to be close together, where X and Y have similar (if not identical) values.  So, I went roughly for the middle of the range, and made my numbers different.  I had visions of the grid forming a line from 0,0 to 65,000+, more densely populated near the origin (0,0) than the outer end (65536,65536), but I  had no quick means of telling whether this was true.

Some time after, with our regions in place, curiosity overcame me and I decided to plot all those regions on a graph.  I used the OpenOffice Calc spreadsheet, with its built-in graphing function, and here is what I get now when I repeat the exercise:

 

Click for full-sized version

Click for full-sized version

 

As you can see, my hunch was right.  The vast majority of the regions (each represented here by a little square) are not only near or on the line x=y, they’re also near its beginning, in the lower numbers.  In fact, of the 1,822 regions, only handful of dots lie both off the predicted line and greater than 15,000 in either axis.

In fact, only three clusters lie significantly off the line – and two of those are ours (I separated the group of 11 into two).  I’m by no means the biggest brain on the grid, nor the most mathematically able; I just happened to notice the effect and avoid it.  If I hadn’t happened to think about it, it’s very likely I’d be on that line with everyone else.  And even being somewhat aware of the effect, I still didn’t do very well – our regions don’t lie as far away from the line as I’d expected.

We can, of course, discount very many of the dots in clusters; these would have been deliberately picked so as to have neighbours.  But it’s clear from the distribution that is there that a sizeable proportion who, like myself, simply wanted a free spot somewhere, anywhere on the grid.

And that’s worth emphasising – this doesn’t tell us anything about anyone’s competence in particular, but it’s a clear illustration of how human beings collectively have difficulty envisaging large numbers.  And the numbers we’re dealing with here, at least in respect of the axes themselves, are hardly geological in scale.

OSGrid aerial image

UPDATE 4th November 2010: I have repeated the exercise. Follow the link below to see the new results:

OSGrid coordinates revisited >>

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One Response

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  1. BlackWolfe said, on October 23, 2009 at 11:19 am

    Yes, this is a new response to a relatively old article… Was searching for “Life Imitates XKCD” on Google just to see what cropped up, and your blog post about this article was among the results.

    Honestly, I’d’ve expected clustering, but not all along X=Y… (I only ever use OpenSim to test things before taking them to Second Life, so I only have the one locally hosted default island and never check for other servers.) Honestly, I’d’ve expected along Y=0, X=0, and abs(x-y)=65536 as well…


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