Poll: The Results

I have been planning to review the results of the polls on my blog for some time, and it seems I have finally gotten around to it. Needlesstosay the turnout has been rather poor. Which is surprising, because 100% of respondents said they vote in polls. Extrapolating that result shows that everybody votes in polls. I guess just not on my blog. Here are the highlights of what we have learnt:

• Judas Iscariot is the people's favourite disciple, however he only got 50% of the vote and doesn't have an overall majority. He needs to form a coalition with any of Simon Peter, James or Bartholomew.
• Amy Pond is the best
• Ainsley Harriot is the sexiest TV chef.
• Robots are our masters
• All sorts of people like cryptic crosswords.
• Absence makes the heart grow fonder

AH was delighted to hear he was voted sexiest chef on TV.

There will now be a change of tack with regards to the polls. I think instead of trying to garner people's opinions on topics I shall ask simple questions with factual answers. You have no reason not to vote now.

Solar Panels

Here is a quick thing that has pissed me off.

I just saw this story on the TV about something to do with financial incentives to install solar panels. I could go back and read it properly, but I doubt I will. While watching, I think, "great, there is a scheme to encourage people to get solar panels, and consequently save energy [and the environment]". But the story quickly turns out to be about how some people are worried about the future of the [solar panel installing] industry. Oh no, some people might loose their jobs! *gasp* Surely they are missing the bigger picture? All those suits they were interviewing should be putting saving the planet as their highest priority and not making money.

Look, here is a house with some solar panels on the roof.

In other news, my exciting TV schedule seems to be winding down. Congratulations to Ash Mair on winning Masterchef: The Professionals. Of course the real winner is the food (or Michel's beard). Greg makes some hilarious noises and Michel makes some hilarious expressions; it really is entertaining viewing. Also congratulations to Harry Judd and his professional partner Aliona Vilani. I would have liked to see Jason Donovan win, but Harry was probably the better dancer.

There has been some sort of comedy show on Saturday evenings. In between Brucie's stand-up slots they play music. There may be some dancing or something going on, but I don't pay that much attention.

This means I have more time for other things. Like being bored. Or maybe, just maybe, working on my blog. I an idea for a series (two) of posts. We will have to see...

Polar Bear Christmas

Can somebody please explain to me what polar bears have to do with Christmas. I totally get that in parts of the northern hemisphere the celebration of Christ's birth happens to fall in the season of winter. So it is natural to associate wintry things with Christmas. E.g. snow, robins and holly with red berries. (Of course the holly is there all year round, but the berries only come out in winter)

What I don't get is polar bears. They don't migrate south into New York state in winter, or anywhere else. So why do we see so many pictures of them cheesing it up for X-mas?

If you want to put this lovable creature in your garden this Christmas then you are a deranged maniac. But the good news is you can totally buy it from Walmart. (Out of stock at time of writing.) Not only that, but America's friendliest supermarket has generously discounted it from $54.00 to $53.93. That is a hugemongous 0.13% saving, after all, this is the season of good will.

Additive Number Theory is Hard

.. for example nobody has published a (correct) proof of the Goldbach conjecture that claims every even number is the sum of two primes. Maybe this isn't actually additive number theory as it concerns the addition of prime numbers, and prime numbers are the building blocks of multiplicative number theory.

But let me tell you a couple of interesting results I found recently on the internet.

Question:
Suppose I want to add some numbers and get a specified total. How many ways are there to do this?

Let's see how many ways there are to add up to 4. You have 1+1+1+1, 1+1+2, 1+3, 2+2 and 4. That makes five ways to add up to 4. Amazing. It makes sense to restrict ourselves to using positive integers, and to ignore the order of the summands. Adding up numbers like this to get a total of n is called a partition of n. So we get to say there are 5 partitions of 4.

Let's define a function!

p(n) = The number of partitions of n.

Simple aye? Here are the first few values. Notice how it grows rather quickly.

n	p(n)
1	1
2	2
3	3
4	5
5	7
6	11
7	15
8	22
9	30
10	42
11	56
12	77
13	101
14	135
15	176
16	231
17	297
17	297
18	385
19	490
20	627

This is the partition function and encodes the answer to the above question. Of course it is still hard to evaluate p(n) for a given n, so we are no closer to answering the above question. But you have to feel that by learning about this function, we are making progress.

If you restrict in some way the numbers you can use to get to your specified total, (e.g. primes or squares) then you can do some interesting maths...

Every positive whole number can be written as the sum of four squares. (The four squares theorem)

Now we know a few technical words I can share the interesting result I found.

If n = 5k+4 then p(n) is divisible by 5.
If n = 7k+5 then p(n) is divisible by 7.
If n = 11k+6 then p(n) is divisible by 11.

It seems crazy to me that the divisibility of the number of partitions of n should be related to n like this. Check them for yourself using the table I helpfully provided. The sequence of numbers in the right hand column is one that naturally turns up all over the place and people love to spot patterns in apparent randomness. I strongly suspect that the ancient Greeks or whoever had seen that these results hold for all the values they could check, but were unable to prove them in general.

These results are attributed to Ramanujan, an Indian mathematician whose genius was so unlike anything before or since that you can't help but regard him as a bit mad. His works are dense with the most complicated formulas and I am always bewildered at how he discovered them.