Lessons taught; Lessons learnt

Last time, we explored how changing the relative values of silver and bronze altered the 2010 Winter Olympic medal table.

This time we concentrate on the top three countries: Canada, Germany, and the USA.

Here is an applet that lets us assign different values to silver and bronze (by moving the blue dot), and so find the scores, and the rankings, of the countries. The initial setting values a silver at 20% of a gold, and a bronze at 20% of a silver. The results, and the dotted curves, are explored below.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

[Source: 2010-top-three places.ggb.]

As you will have found if you played with the applet, the dotted lines indicate where two of the countries swap position in the rankings. Starting on the left (no or low value for silver), the rankings are the IOC ones:

1. Canada
2. Germany
3. USA

As we increase the value of a silver, the first transition swaps Germany and the USA, giving

1. Canada
2. USA
3. Germany

Next, USA and Canada swap places, giving

1. USA
2. Canada
3. Germany

and finally, with silver valued highly, Canada and Germany swap positions, giving the North American-style ranking:

1. USA
2. Germany
3. Canada

Note that none of these options have Germany leading -- the top country is always either Canada or the USA. This answers one of the questions posted in the last post: it is not possible to find any sensible set of values which lead to Germany winning. I say 'sensible' because it is possible to put Germany first if you allocate a negative value to a bronze, but this would not be an option many people would agree with!

The dotted curves are found by setting the 'scores' of pairs of countries equal to each other. If we call 'x' the value of silver relative to gold (from 0 to 1), and 'y' the value of bronze relative to silver (from 0 to 1), then

• Canada = .
• Germany = .
• USA = .

and we have, for example, Canada and the USA's score equal when . Rearranging gives , and the other two curves are derived similarly.

The 2010 Winter Olympics in Vancouver have been over for a few days now. 26 countries out of the 82 that competed won at least one of the 258 medals on offer. Toward the end of the event, quite a lot of interest was generated in which country was going to 'win', in the sense of coming top of the medals table. There's an interesting discrepancy in the medal table rankings around the world, though.

On the NBC Olympics site, and the official Vancouver 2010 site, the top of the medal table looks like this:

On the BBC and on Wikipedia, on the other hand, we see the top of the table looking like this:

The difference comes from a disagreement about the relative value of gold, silver and bronze medals. In the North American sites, all medals are valued equally, and the countries are sorted purely on the total number of medals won. The BBC and Wikipedia, on the other hand, use the standard IOC lexicographical ordering convention: sorting by golds, then by silvers, then by bronzes. This is the ranking system I've always been used to in Olympics, and indeed in any multi-event sporting occasion (such as the World Athletic Championships).

This difference in ranking systems first came to my attention during the 2008 Olympics, when the USA used their ranking system to claim victory over China (China won 51 golds to the USA's 36, but the USA won 110 medals overall, beating China's tally of 100).

One interesting mathematical question behind these tables is this: can we say anything about the rankings which is not sensitive to the particular relative values of gold, silver, and bronze? We might not be able to agree on whether the USA or Canada came first, for example, but perhaps it would always be the case that both of them always rank ahead of Norway. Also, is it possible to find a medal ranking system in which Germany comes first?

Simon Tatham has an interesting writeup of an investigation along these lines for the 2008 Olympic medal tables, which concludes with a wonderful diagram of the partial order of countries (using which we can see that, under any medal ranking system, Great Britain came fourth...). We could generate a similar partial order for the 2010 Winter Olympics.

Rather than doing this, I have created the following applet which will let you explore different medal ranking systems. Simply choose what percentage of a gold a silver should be worth, and what percentage of a silver a bronze should be worth. The applet will calculate the 'gold equivalents' for each country, and display them in their correct rank. The initial setting has been chosen to agree with the official IOC ranking system.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

[Source: Who-Won-the-2010-Olympics.ggb]

Here are some sample results:

Technical details

The above uses some of the Spreadsheet and List features of the new version of Geogebra. If you double click the applet, it should open in its own window, and from there you can open the Spreadsheet and Algebra views and explore how it was constructed.

... or, fun with pentagons:

I run a weekly maths club for a group of primary children in the area, and part of this involves creating or altering resources so that they can be picked up and understood by ten year olds. The first resource from the club that I'm contributing to this blog is my template and instructions for creating a trihexaflexagon.

While there are no shortage of flexagon sites, descriptions and videos around (and I particularly like the Murderous Maths one, as I've seen Kjartan perform, and he was very entertaining), there does seem to be a gap in the market for a simple blank template, together with a simple one-page instruction guide to putting them together.

Source: Inkscape SVG and PDF.

Source: Inkscape SVG and PDF.

I'm happy to work on producing similar instructions for the more complicated flexagons -- leave a comment if you'd be interested.

Having just written about the advantages of self-hosting a Wordpress blog, it would only be right to mention one of the disadvantages: you’re now responsible for the security of the blog.

If you’re like me, then your blog will be something you check on and update only occasionally. It is, however, accessible to the internet 24 hours a day, which means that the opportunity for someone to hack into is it there, whether you are around to look after the site or not.

I’ve recently had someone attempt to hack this site, and redirect visitors to China TV and blueseek (whatever they are!). A quick Google search tells me that I’m not alone. As far as I can tell, all they managed to do was to alter one of the theme files to add a redirect to their website. I’m in the process of removing all the theme and plugin files and replacing them with clean versions… which is one reason why, if you visit this site (rather than use an RSS reader), it currently looks a little more boring than it did yesterday. I will also be sitting down this weekend and going over the file permissions of my entire site with a fine toothcomb… something I suggest everyone else with their own Wordpress blog does.

The message to take from this, I suppose, is never to assume that your personal site will be too unimportant for hackers to bother with.