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Buzz in when you think you know the answer

Aperiodical guest author Andrew Taylor writes about an intriguing piece of number theory which turns out to also be something else.

How many ways are there of writing some natural number $n$ as the sum of two squares?

$$ n = p^2 + q^2 $$

I don’t want an answer for some particular $n$. I don’t even want a general formula. I just want to know… on average.

Meterfy – numbers for everyone!

Fans of numbers will be pleased to hear that they now have their own social network. I’m not sure if I mean than numbers do, or fans of numbers do, but either way Meterfy is a newly launched internet website on which you can share, and discover, a huge quantity of numbers – statistics, constants, totals, averages and molar masses abound.

Almost all above average

This morning James Grime tweeted about a BBC News article, “‘Third of UK postcodes’ have slow broadband speeds“. This quotes Julia Stent, director of telecoms at uSwitch saying:

Britain might be riding the wave of a super-fast broadband revolution, but for 49% who get less than the national average broadband speed, the wave isn’t causing so much a splash as a ripple.

Now, the thrust of the article, that broadband speeds are undesirably slow in some parts of the country, might be valid, but the appeal to the “average” is a very weak argument (Update [23:47]: Although, please see the comment below). The result that 49% are below average should not come as a big surprise!

James, rightly, questions which average is most appropriate, but I am more interested in a tweet by Ian Preston:

We can make almost everyone above average if we are happy for one person to be really badly off.

This, of course, is quite right.

Unless I’m reading it wrong (and I may well be), the Bank of England’s Lending to Individuals December 2011 has outstanding net lending to individuals as £1451.4 billion. The UK Office for National Statistics gives the Public Sector Net Debt excluding financial interventions as £988.7 billion (January 2012) and Total population (UK) as 62.3 million (mid-2010).

If we gave one person all that debt and everyone else zero, then a simple average would be £2.44 trillion divided by 62.3 million people, which is £39,167

Since almost everyone is worth zero, we would almost all be 39 thousand pounds above average. Sound good? This would sort out Government debt and make almost all of us “above average”. And if we’re “above average” then, erm, everything is fine, right?

(Flaws in the argument left as an exercise for the reader!)