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.

Related posts:

1. Who won the 2010 Winter Olympics? An exploration.

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.

Related posts:

1. Who won the 2010 Winter Olympics? The top three places.
2. Geogebra in Wordpress

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3. Midpoints (2): 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.

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2. The Whetstone of Witte online
3. MA Conference: A circular NRICHing activity

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.

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1. Geogebra in Wordpress

Wordpress.com

As far as I can tell, it’s impossible to include Geogebra files in wordpress.com blogs. Wordpress severely restrict the content you can upload to wordpress.com, both for security reasons, and for financial reasons (you can pay them extra to be allowed to embed video/audio). When I tried, it just stripped out the applet information, leaving the ‘please install Java’ text behind.

If you have a wordpress.com blog, then, it looks like you are restricted to hosting your geogebra files on an external system, like Geogebra.org’s own Geogebra upload manager.

Self Hosting Wordpress

Self-hosting means that I’ve got my own Wordpress installation on this site, rather than using wordpress.com, or one of the other free blogging services.

While you might expect that all the wordpress.com restrictions would be lifted on a self-hosted site, this isn’t necessarily true. While there is no issue with using <applet> or <iframe> tags, I recently upgraded this site to Wordpress 2.9 (from the 2.8 version it was running previously), and found that the new version was much more picky about the file-types it allows everyone, even the site admin, to upload. While this is for good security reasons, it was still a little annoying!

There were several ways to deal with this increased security:

1. I could start hosting the Geogebra files on an external site, as above.
2. I could use the new fileless embedding feature which has recently appeared in Geogebra. This lets you embed small Geogebra creations without having to upload a separate .ggb file. Just make sure the ‘.ggb file and .jar files’ option in the Export dialogue is unticked, and the HTML it generates has the file embedding inside the <applet> tag (encoded using Base 64, from the looks of it).
3. I could add .ggb as an allowed filetype to the Wordpress system.

The fileless embedding is useful, but is only sensible for small files — I wouldn’t want to embed a 500kb geogebra file directly into my HTML! For large files I do still want to upload the .ggb files. Rather than hand-editing the Wordpress source, I installed the PJW Mime Config plugin, which lets you edit the allowed filetypes from the standard admin section of the blog.

Other blogs

Another option would be to investigate switching to another blogging platform. For example, it looks like Blogger/Blogspot will let you embed applets in their free service, without too many problems.

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The average of a set of 64 numbers is 64.
The average of the first 36 numbers is 36.
What is the average of the last 28 numbers?

Solution

The total of all 64 numbers is .
The total of the first 36 numbers is .
The total of the last 28 is therefore ,
which makes their average .

Comments

In its ‘raw’ form this is just a basic test that someone understands how to calculate means. It’s nice that the final number is an integer, though… and is it a coincidence that ?

Let’s try the same problem with different numbers:

The average of a set of 51 numbers is 51.
The average of the first 17 numbers is 17.
What is the average of the last 34 numbers?

The average of the remainder is .

Not only do we get a whole number, but the answer is .

Can we conjecture that the answer will always be the sum of the two set sizes? Well, it’s happened twice, so it must be true (!). Looking at the general problem:

The average of a set of  numbers is .
The average of the first  numbers is .
What is the average of the last  numbers?

The average of the remainder is .

Here we might get stuck, unless we remember (or discover, or are taught) at this point the difference of two squares identity: .

This makes our average , and our conjecture is proven.

Extensions

Does a similarly nice thing happen if we specify extra conditions? For example, we could say that there are 5 numbers with an average of 5, and another 9 numbers with an average of 9.

Can we say anything about the median in these situations?

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1. A journey through a mathematical puzzle

(Source: conic_family_plot.ggb.)

Questions and suggestions

Press the play button, and the red curve will animate, showing you some of the possible curves you can get by changing a, b, c and d.

Click 'show controls' and you can alter a, b, c and d yourself. You can also click the checkboxes next to a, b, c or d to see a 'family plot':

These family plots will also animate (and look quite pretty when they do!).

• What are the values of a, b, c and d in the red curve?

  [Answer: at the start, they are all equal to 1.]

• What types of curves would you get for other values of the constants?

  [Answer: The curves you get from these equations are ellipses and hyperbolas.]

The equation is a special case of the equation , which generates all conics.

As well as being a pretty thing to have in my room as a class enters, it could also serve as the basis of investigations into conics (either in the special case shown in the applet, or in general). For example:

• Can we tell by looking at an equation whether it will be an ellipse or a hyperbola?
• Can we tell by looking at the equation whether any points will appear at all, or whether the equation has no solutions?
• Can we classify all the equations which go through one/two/three/more specified points?
• Can we go backwards, from a diagram we want to the equation?
• Why do the family plots look like they do?

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1. Curves and splines for fun and profit
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Incidentally, I've just noticed while creating this post that the new version of Geogebra allows you to embed a simple Geogebra applet completely in HTML, without having to upload a separate .ggb file. A wonderful advance, but it does make it impossible to save the applet to your own computer. I would love to link to the .ggb file here, but the new version of Wordpress seems to have implemented some odd 'security guidelines' for uploads that I need to hunt down and disable!

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1. Interactive Histograms with Geogebra
2. Geogebra in Wordpress
3. Visualisation: Six moving lines

Check with Google if you don’t believe me!

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