Maths 02

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Today I am going to present an interesting photo that appears very confusing: a “proof” that π = 4. Comments on why it is wrong and I will reveal it in a future post.

Thanks to Haggis for tweeting it, and to whoever came up with the photo.

Enjoy. Peace

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Poker 01

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One of the strangest games in the world is Texas Hold’em Poker (the most common and popular variant as played throughout the world). While the principals are easy to explain (deal, bet, check, flop, turn, river – the technical terms if you like) and doing things like ranking the quality of the poker hands, the one things which disturbs me most about poker is that a really bad hand can win and I will be doing a blog on bad beats later.

For now though I thought I would share a basic strategy which I would encourage new/beginner players to adopt; it is adapted from Phil Hellmuth’s very successful “How to play poker like the pros” book which you can buy here. It is an excellent book – and was my first poker book.

Now, on to the strategy: this can be split into early and late (in terms of position). In early position only play the top hands: AA, KK, QQ, JJ, 1010, AK and AQ. In later position, extend the range of hands to 99, 88, 77, 66, AJ, KQ. Finally, always play your blinds if there have been no raises, and only defend your big blind if there has been one raiser and you can get heads up (you want to defend your blind so people don’t get an idea that you can be bullied).

And that is it: this strategy is great for learning the game. It will get you to some final tables and it will ensure that your play is respected over time – people will think of you as a rock (I have this reputation where I play and I find it very useful sometimes).

There are many other facets to the game that we will look at over the coming weeks, but there is no doubt that beginners need to play tight to learn. So next time I will look at some hands that I have played – one final tip; only use this strategy in tournaments – cash games are a whole other ball game and this strategy won’t pay in the long term when playing cash.

Until next time, peace.

End of first week

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Well it has been an interesting first week since I joined online. I have had quite a few visitors to the site (thanks!) but no comments yet – I am sure they will be forthcoming. Before I review the past week, just to update you on the coming week: there will be two blogs – one on poker (Texas Hold’em variant) and one on Maths (I went to the magic shop!)

This week started off back at work and, in fact, there was a lot to do. We had out first Parents evening of the year which went quite well, then this Wednesday some people were taking their Oxbridge pre-interview tests (such as BMAT, Maths test etc.). I hope they did well (certainly the Maths students were well prepared, but they didn’t seem too confident afterwards – as I said to them, it is in the hands of the fates now but they should keep the faith as they won’t be rejected outright unless they scored spectacularly badly and since they are talented I am sure they will be fine!).

Also, I played poker on Thursday and Saturday. While I did not do too well on Thursday, a friend of mine came second in the tournament that day, taking home a few hundred quid, and on Saturday I came 8th in the tournament which meant I got my money back; about sixty quid. I could have gone to a quiz (and probably should have as I saw someone today who said they wished I had been there to answer some Maths questions) but poker seemed more interesting. More on my poker exploits in a blog later this week.

This Sunday I was in work helping to audition some acts for a “talent” night on Tuesday – which turned out to be rather boring as hardly anyone turned up. While I wanted to be Simon Cowell, I never got the chance. So I left at around 2pm-ish and went home to watch the football – and what a game it was! Liverpool 2-0 Chelsea! An excellent result for my team (Liverpool) with Fernando Torres scoring twice; the second a quite sublime shot. I think I screamed out at the TV when it went in! Quality!

This coming week should be a bit quieter – but when I wish for such weeks, it rarely happens. Hey-ho. Peace!

Maths 01

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As promised I want to talk to-day about a little bit of Maths. As a Mathematician, I think that there are some things which more people should know – beyond “Fermat’s Last Theorem” or “Pythagoras'” from what they may have learned at school.

Today I am going to talk a bit about sequences. Here in the UK, most students will learn about arithmetic progressions (things like 1, 4, 7, 10 where the difference between terms is constant) and geometric progressions (things like 1, 3, 9, 27 where the ratio between terms is constant) in any A-Level course. These have related series (where we add up terms of the sequences). And hopefully most students will have learned the formulas for each. One area where pupils seem to get confused is when we start adding up an infinite number of terms of a sequence – infinite series are very weird things indeed!

There are also other types of sequences, such as alternating sequences (where the terms are, as the title suggests, alternating their sign) and let us look at the following alternating series.

An example:
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – … (where the … mean continuing on for ever)

Well what does this look like? What is the sum if we assume there are infinitely many terms?

We can consider the partial sums – that is, the finite sums, which becomes 1, 0, 1, 0, 1, 0, 1, 0 etc. so if would appear that the answer has to be 1 or 0 depending on how many terms we are adding. But infinity is not a number – it is a concept (again there will probably be a similar blog post on this confusion in the coming months!).

Well, we can pair up the terms as follows

(1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + … where each bracket is 0 and since there will be infinitely many zeros being added up the sum is 0 (or is it?)

Instead we could have paired up the terms as follows

1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) + … where each bracket is 0 and since there are infinitely many zeros being added then the sum is 1 (since we paired up from the second term rather than the first term)

So which is correct? Well, unsurprisingly, it depends on your definition; often in Mathematics it will depend on the context and here we have not been specific enough to say which is “correct” in the usual sense.

Let us assume that the sum exists, call it k. Well k = 1 – 1 + 1 – 1 + 1 – 1 + … and notice that we can then write

k = 1 – (1 – 1 + 1 – 1 + 1 – 1 + 1 – …) so k = 1 – k which is a simple equation 2k = 1 so k = 1/2

ARGH! So the sum, k, is 1/2. This certainly, on the face of it, seems more logical as we have only done a trick similar to that which shows 0.999… = 1 (see here) where we helped to eliminate the infinite string of 9s with a little mathematical magic. And in fact, this is the “accepted” sum. So if you add up infinitely many 1s and -1s then you obtain 1/2.

This series if called Grandi’s Series and has a rich history.

I hope you enjoyed this little introduction to some Mathematical magic which certainly seems to upset the usual way of thinking! Next time we will extend on some of these ideas and show something even more, apparantly, ludicrous (and explain why it makes sense) that

1 + 2 + 3 + 4 + 5 + … = -1/12

That will require Magic so I had better go shopping. Peace.

Back to work

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Today (Monday 1st November) we go back to work. It should be an exciting half-term in the run up to Christmas – 6 weeks of work left, and then a holiday from December 10th until the start of January – as most of the work will be done, particularly for those who have examinations in January.

This half-term has been a bit of a peculiar one – I went away for a few days in Edinburgh to visit my friend Haggis (you can follow Haggis here) – and then was back here for a few days and everything seems to have been a bit of a blur. Haggis is also a Mathematician and works in knot theory (something which I touched on at university, but which I am certainly nowhere near being expert – or competent – in).

One thing my first Mathematicial blog will be on that Haggis reminded me of (on twitter) is about making infinite sums make sense. I am sure you will look forward to that.

Another thing is that I am breathing a lot better now that my owner has quit getting 20-30 fixes per day.

Finally, it is World Series time at the moment – I was introduced to baseball in about 2002 by my research supervisor, and the San Francisco giants in particular. Well, its 2010 and the Giants are again in the “Fall Classic” and, even as I write this blog post, they are about to start Game 4 (holding a 2-1 lead at present, with the prospect of facing Cliff Lee again in Game 5 – the Texas star pitcher). I have followed the Giants ever since and, during the Bonds era always hoped they would win for him. However, they have a chance now without Barry, and how remarkable it would be for them as they have not won a series since they moved out to the West Coast. Finger crossed.

I will be posting my first Mathematical blog on Tuesday so until then, peace 🙂

Hello world!

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First post: Happy Halloween!