This is the first of a couple of posts on the mathematical approach known as the Monte Carlo method. I want to start gently by taking a (virtual) trip to Monte Carlo in Monaco and its famous casino, specifically to explore a once well known phrase - the man who broke the bank at Monte Carlo. The article below is extracted from my book Dice World . A roulette wheel is a physical device, and as such is not a perfect mechanism for producing a random number between 1 and 37 (or 38 in the more money-grabbing US casinos). Although wheels are routinely tested, it is entirely possible for one to have a slight bias – and just occasionally this can result in a chance for players to make a bundle. It certainly did so for 19th-century British engineer Joseph Jagger, who has, probably incorrectly, been associated with the song ‘The Man Who Broke the Bank at Monte Carlo’, which came out around the same time as Jagger had a remarkable win in Monaco. The song probably referred instead to the conman Ch...

It was interesting to see in today's paper that a neuroscientist is chiding teachers for preventing children from counting on their fingers. The practice is apparently frowned on because it is childish and it was assumed that it prevents internalisation of the numerical processing. Professor Jo Boaler of Stanford University is quoted as saying 'Teachers are stopping children using their fingers at a ridiculous age - four or five - so that has to change.' She point out that when we work something out mathematically, the brain maps this onto fingers - and better maths achievement goes hand-in-hand with better finger perception. I think she possibly stretches this a little too far by saying 'It explains why musicians, particularly pianists, typically have a higher level of understanding of mathematics' - but apart from anything else, to discourage the use of a readily available resource seems crazy. I am happy to admit that if I am asked by a website to input the...

String theory is something that I've been highly sceptical about for some time, influenced by books like Not Even Wrong and The Trouble with Physics . This meant that a recent book, Why String Theory? by Joseph Conlon has proved a very interesting read to provide an explanation for the popularity of string theory among physicists, despite its apparent inability to make predictions about the real world. I can't say the new book has won me over (and I ought to stress that, like Not Even Wrong , it's not an easy read), but what I do now understand is the puzzle many onlookers face as to how physicists can end up in what appears to be such an abstruse and disconnected mathematical world to be able to insist with a straight face and counter to all observation that we need at least 10 and probably 11 dimensions to make the universe work. It seems that string theory emerged from an attempt to explain the strong force back in the late sixties, early seventies. The idea ...

Image based on NASA image , credit ESA/NASA/SOHO As someone who writes about physics and cosmology I occasionally get asked my opinion on something like the black hole firewall paradox. If I'm brutally honest (which I rarely am, because I'm far too polite) I will reply: 'I don't know. I don't care. It bores me stiff.' In case you aren't sure what the paradox is, it emerges from a combination of quantum theory and general relativity (which don't go together, but hey), and relies on piling about four levels of mathematical supposition on top of each other to come to the conclusion that the information that could be considered to exist on the event horizon of a black hole can't (as it was hypothesised it did) represent all the information in the 3D interior with gravity included, and 'therefore' something passing through the event horizon would burn up. Simples. This topic involves theorising about a phenomenon that almost certainly ...

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3. Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable. It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5.  However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign. Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only wor...

In the book I'm writing at the moment I'm considering the relationship of the arrow of time to entropy, the measure of the disorder in a system that comes out of the second law of thermodynamics. Entropy can be calculated by looking at the number of different ways to arrange the components that make up a system. The more ways there are to arrange them, the greater the entropy. As an example of why this is the case, I was talking about the letters that go together to make up that book, and the very specific arrangement of them required to be that actual book. Assuming that there will be about 500,000 characters including spaces in the book by the time it's finished, then there are 500,000! ways of arranging those characters. That's 500,000 factorial, which is 500,000x499,999x499,998x499,997... - rather a big number. It's not practical to calculate the number exactly, but there are approximation techniques, and if the large factorial online calculator I found ...

Okay, here's a word association test. What's the first thing that comes to mind when I say... mathematicians ? Hands up how many of you said 'Fun'? What, no one? If you are a mathematician, or a physicist making heavy use of maths, you may feel there's plenty of fun in your world, but just in case you needed a bit more, I can highly recommend UCL's new e-magazine for mathy people, Chalkdust . (Rather an odd choice of title - a bit like a computing magazine calling itself Abacus . But we are dealing with mathematicians.) What I ought to say straight away is that Chalkdust (my spellchecker insists on converting that to  Chalkiest ) is not a magazine version of an Ian Stewart type, light and fluffy popular maths book. This is a magazine that doesn't shy away from including the equations of general relativity. But having said that, you don't have to be a genuine, heavy duty mathematician to get something out of it. When I was at university, my maths ...

Infinity, as no end of people keep telling me since I wrote A Brief History of Infinity is a big subject, so I like to revisit it now and again. One of the joys of doing my talk on infinity, a real favourite of mine, is the way people's minds are duly boggled by the idea that there can be something bigger than infinity. And what's more, you can prove it without a single equation. Thanks to the great German mathematician Georg Cantor we can establish this painlessly. The first step is to discover the concept of cardinality in set theory. A set is just a collection of things, and set theory is the maths that describes the workings of such collections, and from which all the basics of arithmetic can be derived. Cardinality is a measure of the size of the set, and the important thing to be aware of is that if we can pair off items in two sets so they are in one-to-one correspondence, those sets have the same cardinality - they are the same size. Take a simple example - legs on...

I'm delighted to say that my latest book, Dice World is now available for sale. Subtitled 'science and life in a random universe', it's about randomness (well, duh), probability and statistics. It explores how the ‘clockwork universe’ imagined by Newton, in which everything could be predicted given enough data, was disproved bit by bit, to be supplanted by chaos theory and quantum physics. This is a world in which not only is accurate forecasting often impossible but probability is the only way for us to understand the fundamental nature of things. Where else do you get a chance to meet Maxwell's Demon, Schödinger's cat and take part in an experiment using Bayesian statistics to see how a mug on my desk alters the probability of my owning a golden retriever (no, really)? I've really enjoyed writing this book, and I hope it will be of wide interest. If you fancy buying it, it would be ideal if you could use the links below ( or from my website ). (Apolog...

These are twins. The one on our left is older. I have had an interesting discussion with Paul Nahin, the author of The Logician and the Engineer , which I'm currently reading to review . Nahin quotes a logic problem that is apparently well known amongst mathematicians. In it, one person is trying to guess the (integer) ages of the other's three daughters. He is given some information that allows him to narrow the possible ages down 1, 6 and 6 or 2, 2, and  9. Then the first gives an additional pieces of information. 'My oldest daughter,' he says, 'likes bananas.' Immediately the second person knows the girls' ages. The accepted correct solution goes that the daughters can't be 1, 6 and 6 because there isn't an oldest daughter in this scenario, so our logician can deduce they are 2, 2 and 9. But I say that this is rubbish - at the very least poor logic. Why? It is perfectly possible to have two six-year-old daughters born 10 months apa...

There is a Turing statue in Manchester, but frankly it's unrecognisable. You can do better, guys. There is nothing editors like more than anniversaries. Recently I suggested a feature to a magazine. 'It could work,' they said, 'as long as you can find an anniversary to tie it to. We need a hook.' Frankly, this is a load of rubbish. The reading public really doesn't care why a magazine or newspaper is coming up with a particular story as long as it's interesting. But editors feel they have to devise a justification. They need a reason that a particular story should be used, so they arbitrarily use the factor of a significant date. It keeps them happy, bless them. This being the case, we can expect a flood of books on Alan Turing as it was the 100th anniversary (wey-hey!) of his birth in June. Leaving aside the fact Turing would certainly have preferred a binary anniversary (2018 will be the 1000000th anniversary of his death), I'm currently reading...

Thanks to Peet Morris for this excellent example of probability running counter to common sense. Imagine I have a huge stack of coins and flip them one after another. These are fair coins, with a 50:50 chance of coming up heads or tails. First of all I flip the coins one after another (leaving the flipped coins on the table) until the sequence H T H comes up. At that point I stop and count the coins. Then I repeat this experiment many times. For the second part I again flip the coins, leaving them on the table, until the sequence H T T comes up. At that point I stop and count the coins. Then I repeat the experiment many times. On average would you expect it to take more flips to produce H T H, more flips to produce H T T or the same number of flips? Common sense says this is pretty obvious. It's the same number of flips. And certainly if I take three coins and flip them, there's the same chance of H T H or H T T coming up. But, remarkably, things are different in the...

In my teens I was fascinated with a mathematical construct called a tesseract. This is a four dimensional hypercube. A cube is constructed in three dimensions from six faces, each a square. A tesseract is constructed in four dimensions from eight faces, each a cube. Funnily both my youthful introductions to tesseracts were from fiction. The first was in Madeleine L'Engle's wonderful children's classic (still very readable for adults) A Wrinkle in Time . I know some people don't like this book because it has an underlying religious message, but for me it is one of the best children's science fiction stories ever. In it, a tesseract is a gateway for interstellar travel. But much more informative is Robert Heinlein's brilliant short story, And he Built a Crooked House , which appears in the collection The Unpleasant Profession of Jonathan Hoag . Dating back to 1941, this story tells of an architect who builds a tesseract house. Now, clearly you can't build ...

If I'm honest, there are probably not many children who, when asked what they want to be when they grow up, say 'A mathematician.' Similarly, while many kids want to be famous, have a hit record or play sport for their country, not too many, when asked, would admit that they would want to have a mathematical theorem named after them. But when you're a bit older, you have to admit it's kind of cool. After all, it didn't do Fermat or Pythagoras any harm. Well, now you can have a theorem named after you - without doing the hard work. A spin-off of the University of Edinburgh, TheoryMine , produces new, unique mathematical theorems which for a small fee (£15) they will assign to you, allowing you to give it whatever name you like. You might object that, since you didn't devise the theorem, your name can't be attached to it. Well, we don't know if Fermat did devise his 'last theorem' and Pythagoras definitely didn't originate his - and the...

Okay it's time to don the grumpy old man suit and make the neighbourhood safe for humanity. Listening to the radio the other day I found myself cringing at something that has always got my back up when talking to primary school teachers about maths. These days if you want to do, say, multiplication, there are a number of different techniques available. The teachers refer to these (often while speaking to baffled parents) as 'strategies.' They may even ask little Johnny 'Which strategy are you going to use, little Johnny?' No, no, no, NO! These are not strategies. The strategy is having a range of different techniques. A strategy is a broad direction, not a specific methodology. The specific approach being employed at any one time is a tactic, or a technique or a method. It is not a strategy. It is not strategic, it is tactical. Why do they do this? We've got perfectly good English words for what they want to say, but they have to distort the meaning of ...

Generally speaking I'm a bit so-so about recreational mathematics . I can't get very excited about polyominoes or tiling, for instance. But when the field strays into probability I get fascinated - and the mind gets boggled. Take the little probability problem mentioned in the New Scientist article I've linked to there. It gets rather lost in the article, and they don't describe it particularly well. Let's take a look. The problem statement is simple. I have two children. One is a boy born on a Tuesday. What is the probability I have two boys? But to get a grip on this problem we need first to take a step back and look at a more basic problem. I have two children. One is a boy. What is the probability I have two boys? A knee-jerk reaction to this is to think 'One's a boy - the other can either be a boy or a girl. So there's a 50:50 chance that the other is a boy. The probability that there are two boys is 50%.' Unfortunately that's wrong....

Of all the talks I do in schools, public events and businesses , the one I enjoy most is the one on infinity . There's just something wonderful about the mix of fascination and sheer boggledness of mind that I see in the faces and get from the feedback afterwards. This boggling is nothing new. The first person to really consider the true mathematical oddities of infinity in any depth was Galileo. In his book Discorsi e dimostrationi matematiche, intorno a due nuove scienze (Discourses and Mathematical Demonstrations Concerning Two New Sciences) he explores infinity in a way that had never been done before. This book is his masterpiece. Forget the one about the Earth going round the Sun that got him into such trouble, this book sets up the basis for mechanics, laws of motion, relativity and more. Written while he was under house arrest, it's in the form of a discussion between three characters. These are Filipo Salviati (named after a friend of Galileo’s who died 16 years b...