Marathon runners need never “hit the wall” again thanks to a mathematical model that will help them reach the finish line in their best time.

More than 40 per cent of marathon runners will hit the wall during a race, experiencing sudden pain and fatigue as their carbohydrate reserves run low and their body switches from burning carbohydrate to burning fat. So Benjamin Rapoport at the Massachusetts Institute of Technology has given runners an online calculator that will tell them how much carbohydrate they need to consume to have enough for a whole race.

Read the rest at New Scientist.

It has taken 15 years to get to this point, but it is now clear that every possible scrambled arrangement of the Rubik’s cube can be solved in a maximum of 20 moves – and you don’t even have to take the stickers off.

That’s according to a team who combined the computing might of Google with some clever mathematical insights to check all 43 quintillion possible jumbled positions the cube can take. Their feat solves the biggest remaining puzzle presented by the Rubik’s cube.

“The primary breakthrough was figuring out a way to solve so many positions, all at once, at such a fast rate,” says Tomas Rokicki, a programmer from Palo Alto, California, who has spent 15 years searching for the minimum number of moves guaranteed to solve any configuration of the Rubik’s cube.

Read the rest at New Scientist.

It’s been a while since I covered this kind of dodgy maths, so let’s go over the basics. “Formula for” stories are seen by PR agencies as a great way to get free press coverage for whatever product they are shilling because the equations can be dressed up as real research. Attaching a “Dr” or “Prof” to your news story is a great way to gain legitimacy, and the media lap it up as another example of what those crazy boffins are up to.

While this is all great for the PR agencies and their clients, it’s terrible for science. These formulas tend to be based on extremely dodgy assumptions and contain variables which can’t be objectively measured. What’s worse, even a simple mathematical analysis usually reveals problems such as division by zero, which can lead to things like cold and lumpy but infinitely perfect pancakes.

With these problems in mind, let’s take a look at the formula for the perfect handshake. It was created by Geoff Beattie, head of Psychological Sciences at the University of Manchester, and is detailed in this the press release:

I’ve broken it over two lines because the thing is so long, and I think that square root is meant to cover the entire equation, not just the first term, but the press release isn’t very clear. We’ve also got a definition for the many variables, along with what I assume is their optimal values:

(e) is eye contact (1=none; 5=direct) 5;
(ve) is verbal greeting (1=totally inappropriate; 5=totally appropriate) 5;
(d) is Duchenne smile – smiling in eyes and mouth, plus symmetry on both sides of face, and slower offset (1=totally non-Duchenne smile (false smile); 5=totally Duchenne) 5;
(cg) completeness of grip (1=very incomplete; 5=full) 5;
(dr) is dryness of hand (1=damp; 5=dry) 4;
(s) is strength (1= weak; 5=strong) 3;
(p) is position of hand (1=back towards own body; 5=other person’s bodily zone) 3;
(vi) is vigour (1=too low/too high; 5=mid) 3;
(t) is temperature of hands (1=too cold/too hot; 5=mid) 3;
(te) is texture of hands (5=mid; 1=too rough/too smooth) 3;
(c) is control (1=low; 5=high) 3;
(du) is duration (1= brief; 5=long) 3.

Both the formula and its variables are looking really dodgy. I’ve literally no idea what terms like {(4<c>2)(4<du>2)}² are meant to mean. I can only think that the angular brackets denote some kind of average, but then why do they only apply to some of the variables? Are those 2s actually meant to be ²? In which case you can rewrite the whole term as (2<c><du>)⁴, which is at least a little bit simpler.

I also take issue with using two letters to stand in for one variable, because they can be confused for two separate variables multiplied together. Measuring “verbal greeting” and “vigour” doesn’t mean that both of your variables have to start with a v – real mathematical equations make extensive use of Greek letters in an effort to solve this exact problem. But even if this equation was beautifully formatted, it would still be rubbish.

All the measurements are completely subjective, and the scales of 1 to 5 indicate the data behind the equation was probably collected from a survey. This even includes variables such as temperature, which can easily be measured scientifically. Remember, subjective measurements are one of the hallmarks of a “formula for”.

I emailed Beattie yesterday to ask how the formula was created, but as he is yet to reply I can only speculate. I think what he has done is ask people a bunch of questions about handshakes, and then tried to fit their answers to some kind of least-squares model, as indicated by the squares and square root in the formula. This method gives you a great equation for “explaining” the data you’ve gathered, but doesn’t necessarily tell you anything about the phenomena you’re examining.

If that is the case, I still don’t understand how the formula is meant to work. You’d expect that the perfect handshake would have a maximum value of PH, and since there is no division or subtraction involve, that just means slotting in the maximum values for all your variables. The optimal values in the press release include a few 3s and 4s though, so PH isn’t going to be maximum. Hmm.

As with all “formula for” stories the maths behind the perfect handshake formula just doesn’t add up, yet it’s being interpreted as a serious piece of research. Comments on the Mail story such as these two show just how much damage this can do to people’s impressions of science:

“So, most of the country is out of work desperately trying to survive and these idiots are getting paid, what – to study handshakes? Sack these people immediately!”

How much time did the nutty professor spend on this useless bit of information?

Mathematical models and equations are a fantastical tool for understanding the natural world around us, but they have to be based on sound assumptions and decent science – things that “formula for” stories such as this almost invariably lack.

In the past I’ve linked to all kinds of periodic tables, from the edible to the audiovisual. Now, someone’s gone all meta and created a periodic table to list all of these periodic tables:

You can see a larger version here, complete with links to all the other tables.

And you think your job is tough…

Popular Science has drawn up a list of the ten worst jobs in science, which includes thankless tasks such as “armpit detective” and “whale slasher”. Don’t let them put you off pursing a career in science however, as the list also reveals the best job: “multispecies baby tickler”. Where do I sign up?

Fire! De der deeeer, der der…

A Ruben’s tube is a nifty demonstration of standing waves with a healthy dose of burnination:

A really geeky maths joke

I probably find this joke far more amusing than I should:

An engineer, a physicist and a mathematician find themselves in an anecdote, indeed an anecdote quite similar to many that you have no doubt already heard.

After some observations and rough calculations the engineer realizes the situation and starts laughing.

A few minutes later the physicist understands too and chuckles to himself happily as he now has enough experimental evidence to publish a paper.

This leaves the mathematician somewhat perplexed, as he had observed right away that he was the subject of an anecdote, and deduced quite rapidly the presence of humour from similar anecdotes, but considers this anecdote to be too trivial a corollary to be significant, let alone funny.

Here’s an extract from an article I wrote for New Scientist in honour of Pi Day today.

The stars overhead inspired the ancient Greeks, but they probably never used them to calculate pi. Robert Matthews of the University of Aston in Birmingham, UK, combined astronomical data with number theory to do just that.

Matthews used the fact that for any large collection of random numbers, the probability that any two have no common factor is 6/pi2. Numbers have a common factor if they are divisible by the same number, not including 1. For example, 4 and 15 have no common factors, but 12 and 15 have the common factor 3.

Matthews calculated the angular distance between the 100 brightest stars in the sky and turned them into 1 million pairs of random numbers, around 61 per cent of which had no common factors. He got a value for pi of 3.12772, which is about 99.6 per cent correct.

A serious science survey?

The BBC reports that one in 10 children believe the Queen invented the telephone, while others suggest Charles Darwin and Noel Edmonds. The results come from a survey of 1,000 school kids, but rather than despairing at the state of science education, I’m actually amused by this story.

These types of articles seem to crop up fairly often, with children giving nonsensical answers to questions about historical facts. Everyone always interrupts them fairly seriously, but I think it’s far more likely that the kids are just having a laugh.

High-gravity lava lamps

Would a lava lamp work on Jupiter? There’s only one way to find out – build a giant, semi-lethal centrifuge out of Meccano, and take your lamp for a spin:

I wrote this piece for the Guardian as part of their Valentine’s Day coverage:

Steamy love poems are always popular around Valentine’s Day, but can a few lines of tender verse really make people hot under the collar? Researchers at Aberystwyth University attempted to find out earlier this week, using thermal imaging cameras to take the temperature of volunteers reading the work of Romantic poets.

The experiment is a collaboration between the arts and the sciences, led by poet Richard Marggraf Turley from the Department of English and Creative Writing and Reyer Zwiggelaar from Computer Science. They asked six volunteers from each department to silently read 12 love poems, while a slightly less amorous text about thermal imaging served as a control. As the participants pored over poems, including Bright Star by John Keats and To His Coy Mistress by Andrew Marvell (both are reproduced in full below), thermal cameras monitored their faces for any change in temperature that could reveal their true feelings.

Read the rest at the Guardian.

A “new” formula for marriage? Not quite

A number of news outlets have run stories on a formula for finding your “Optimal Proposal Age”, based on a press release from the University of New South Wales. Far from being a new result, it’s actually a repackaging of an old mathematical puzzle known by a variety of names, including the marriage problem.

Imagine you’ve decided to search for the perfect partner by going on 100 blind dates. After each date you decide whether you want to marry the potential suitor, and if you choose not too you can never see them again. Contrived, but then this is a maths puzzle!

How do you pick your partner? If you wait until the end of all 100 dates, you’ll be stuck with whoever is on the end of the list, whether you like them or not, but if just go for the first person you like then you could be missing out on someone who is a better match. It turns out that the best strategy is to see the first 37 potentials, then pick the next one who is better than those 37. Not the most romantic approach, but at least it makes for a quirky Valentine’s Day news story I suppose.

The Mail and others have leapt on a rather silly paper by Peter Backus, a University of Warwick economist. He’s used the Drake equation, which was originally intended to estimate the number of alien civilizations in our galaxy, to explain why he doesn’t have a girlfriend.

You can visit Wikipedia for an explanation of the Drake equation, or alternatively check out Colin’s dissertation for the full details. The basic idea is to break down all the requirements for alien life in to individual factors, such as the chance of a star having planets or a planet supporting life, then multiply them together to get the number of civilizations out there in space. Trouble is, we don’t have very reliable evidence to back up most of the figures, so estimates vary wildly.

Backus has used the same principal to find his perfect woman, and “discovered” that there are only 26 women in the UK that are suitable for him. That’s a one in 285,000 chance of meeting “the one”, apparently. Of course, the exact same criticism of the Drake equation can be applied here – most of his numbers are entirely subjective and not backed up by evidence. Pick some different numbers, and you’ll come up with an entirely different answer.

I can’t really blame Backus for his formula, as it’s not like he’s trying to sell anything or has got the maths wrong. What I find annoying is the way the media leaps on the figure of “one in 285,000″ as an absolute fact, and describes maths no more complicated than multiplication as if it were some sort of advanced calculus that should only be attempted by a genius. Let’s just hope no one discovers the ancient art of “division”, or our heads just might explode.

I actually had some success, and after more than 12 hours of work was rewarded with this rather understated error message:

Quite. Sadly, in the end I had to say goodbye to my faithful old PC and buy a new one, complete with Microsoft’s latest operating system, Windows 7. It’s quite different to the Windows XP I’m used to, especially as I’d disabled most of XP’s bells and whistles to make it run like Windows 2000. Essentially, I’ve been using the same operating system for an entire decade, and now I’ve been forced to change some long-held habits!

All of which leads me on in a fairly rambling way to what I had originally intended to talk about at the start of 2010 – whether we’re now living in a new decade. The media seem pretty convinced that we’ve abandoned the “Noughties” in favour of the “Teens”, but the maths says otherwise – it won’t be until the end of 2010 and the start of 2011 that we enter the next decade.

It’s the same argument that you probably tired of in the years leading up to December 31st, 1999. At the time, mathematicians said that millennial celebrations should be put off until the start of 2001, while the rest of the world largely ignored them.

Simply put, our calendar system starts at the year 1 AD, not the year 0 AD. One year later is 2 AD, ten years later is 11 AD, and two-thousand years later is 2001 AD. So, new decades start with years ending in a “1″.

But when we speak of the Noughties, we obviously mean the years 2000 to 2009. The year 2010 can’t be a Noughtie, because it doesn’t have a 0 in the right place. And hang on a moment, isn’t the calendar based off the life of Jesus, a man whose date of birth we know very little about? And lets not even start on the missing 11 days of September 1752.

Given the human desire for patterns and our fondness of round numbers, it’s probably best if we stick to celebrating 2010 as the new decade – it’s no less arbitrary than any other choice. Even so, I can’t help wanting to go with 2011. It may be ugly, but it’s mathematically correct!

A new technique for unpeeling the Earth’s skin and displaying it on a flat surface provides a fresh perspective on geography, making it possible to create maps that string out the continents for easy comparison, or lump together the world’s oceans into one huge mass of water surrounded by coastlines.

“Myriahedral projection” was developed by Jack van Wijk, a computer scientist at the Eindhoven University of Technology in the Netherlands.

“The basic idea is surprisingly simple,” says van Wijk. His algorithms divide the globe’s surface into small polygons that are unfolded into a flat map, just as a cube can be unfolded into six squares.

Cartographers have tried this trick before; van Wijk’s innovation is to up the number of polygons from just a few to thousands. He has coined the word “myriahedral” to describe it, a combination of “myriad” with “polyhedron”, the name for polygonal 3D shapes.

You’ll find the rest at New Scientist, along with some nifty images and video.

If you don’t remember him, Cliff Arnall often pops up in to the media peddling mathematical nonsense. The Mail bill him as “Professor” Arnall, which is a new one, but it’s not entirely clear which institution he’s from. Certainly not Cardiff University, who have made repeated attempts to distance themselves from Arnall after he left their employ as a part-time tutor.

Let’s have some fun playing with the formula then. The Daily Mail have a handy explanation:

All the variables in the left column are basically arbitrary scores out of 5, and thus fairly meaningless. In the right column, T, L and C are at least all quantifiable, in that we can assign a meaningful value to them. Multiplying T by L is actually fine, because both of these variables use units of time. The problems start when you divide by the square root of C.

Quick anyone, what’s the square root of £1? I might as well ask for banana divided by orange – neither question makes mathematical sense, because there is no such thing as the square root of currency.

Our old friends zero and infinity make an appearance as well. If a toy is free, it doesn’t matter if you give it 0 out of 5 for everything else, because as long as your child plays with it for even a second, it’s going to have infinite play value. Dividing by smaller and smaller values of C makes the last term in the equation grow rapidly, completing dwarfing the others. In other words, Cliff Arnall’s perfect toy is crap and worthless. Just like his formulas then.