## The Monty Hall Problem

You’re on a game show and the host asks you to pick one of three doors. Behind one of them is the star prize: a sports car. Behind the other two are goats. Once you have made your pick, the show host opens one of the other doors – always revealing a goat. The host then asks if you are happy with your original choice, or whether would you like to switch.

## The Monty Hall Problem

The Monty Hall Problem is based on a game show called Let’s Make a Deal, originally presented by Monty Hall.

This problem was posed to the mathematical community in a letter to The American Statistician in 1975, but was popularised by Marilyn Vos Savant in 1991.

Marilyn’s article suggested that the right thing to do is to switch. If you read the article – and please do – you will see that Marilyn received many very negative comments – some would say rude,  to her conclusion. Here are some examples:

• You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!
• Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful.
• Maybe women look at math problems differently than men.
• You are the goat!

It is incredulous that people could write such comments and – and it is now accepted that Marilyn was correct and the right thing to do is to switch doors. There would have ben some red faces and I hope they apologised.

## Solution

If you are faced with three doors and choose one of those at random, I think we’d all agree that you have a one in three (1/3) chance of choosing the car.

If one of the other doors is opened, showing a goat, one argument says that switching doors now gives you a 1/2 chance of winning the car.

Another argument says that once you have chosen a door, you have a 1/3 chance of winning the car. If you could choose the other two doors instead you would have a 2/3 chance of winning the car. If you are now showed one of these doors – revealing a goat – by switching, you still have a 2/3 chance of winning the car. Of course, you will not always win a car but over many games, by switching you will win 2/3 of the time, compared with only winning a 1/3 of the time if you do not switch.

There are two ways that might persuade you that switching is the right thing to do.

1. Increase the number of doors to 50. When you choose a door you have a 1/50 chance of winning the car. This means that there is 49/50 chance that the car is behind one of the other doors. The host now opens all the remaining doors, just leaving your original door and one other. If you stick with your original door you still have a 1/50 chance of winning the car. If you switch, you have a 49/50 chance of winning the car.
2. We can run a simulation. In preparing this article, I wrote a Java program to do that. We ran the Monty Hall Problem 5,000 times, both switching doors and not switching. If we have three doors, by not switching, the car was won 1702 times, compared to 3310 when switching. These are close enough to the 1/3 and 2/3 win ratios that we would expect. If we now change the number of boxes to 50, if we don’t swap we win the car 99 times. If we swap, we win 4,914 times. Again, this is close enough to the theoretical figures to persuade us that switching is the correct thing to do.

Hopefully, you are now persuaded that you should switch doors when faced with a similar situation. Good luck!

This post was also published on LinkedIn.

# The future of scientific publishing: let’s make sure it’s fair as well as transparent

Scientific publishing has undergone a revolution in recent years – largely due to the internet. And it shows no sign of letting up as a growing number of countries attempt to ensure that research papers are made freely available. Publishers are struggling to adapt their business models to the new challenges. But it is not just the publishers who struggle.

Peer-reviewed publications are extremely important for academics, who use them to communicate their latest research findings.
When it comes to making decisions about hiring and promotion, universities often use an academic’s publication record. However, the use of publication consultants and increasingly long lists of authors in certain disciplines are changing the game.

So where will it all end?

## Publication consultants

When a scientific paper is published, the authors have an obligation to report who has contributed. This recognition can take the form of authorship, acknowledgements or by citing the work of others. Most publishers will provide details about how to recognise various types of contribution. For example, the Institute of Electrical and Electronics Engineers (see page 14, section 6) says that a statistician helping with analysis, a graphic artist creating images or a colleague reviewing an article before submission should all be recognised in the acknowledgements section of an article.

Wikimedia Commons

However, recent years have seen a growing industry where publication consultants offer to help authors, or even institutions, to get their work published. The consultants charge a fee for this service. The type of help that is available ranges from proof reading, data collection, statistical analysis, helping with the literature review and identifying suitable journals to approach for publication.

We should ask why academics need these kind of services. Surely, institutions already provide this type of support to its less experienced researchers – and more experienced researchers, especially those with a PhD, should be qualified to carry out these activities themselves. After all, carrying out research and writing scientific papers is an essential part of PhD training.

If researchers do feel the need to use the services of a consultant, it should be made transparent either including the consultant as an author on the paper, or at least acknowledging their services – otherwise a prospective employer, a promotion panel or future collaborators can never be sure if there was somebody else helping with the paper. It might also be appropriate for publication consultants to provide an annual return detailing the papers on which they have consulted.

## Growing author lists

To increase the transparency of academic publishing it may therefore seem that adding more people on a paper is the way forward. But there’s also another way of looking at it. Earlier this year, Physical Review Letters set a record when it published a paper with 5,154 authors. Such huge author lists are becoming increasingly common. In most disciplines this would seem excessive and we might ask whether all these authors did contribute to the paper?

Some have argued that this development is threatening the entire system in which academic work is rewarded. So what should we do about it? A radical suggestion could be to remove authors on papers completely and replace them with project names. Another suggestion, already practised by journals such as Plos One, is to list the contribution of each author. Whatever your view, there can be little doubt that some disciplines use different metrics to measure contribution.

## Open Access

The traditional way to publish a scientific article is to submit it to a journal and, if accepted, you sign over the copyright to the publisher. Your article is then sold via institutional subscriptions or individual payment when it is downloaded.

There are problems with this model: a common objection is that the people who do all the work – the authors and reviewers – get no payment and yet the copyright is assigned to a publisher. Worse, the authors, reviewers and taxpayer (who funded the research to start with) then have to pay to read the article. Of course, the publishers do have costs, such as staff, printing, web site maintenance, registering DOI’s etc –and they are typically companies that need to make a profit.

Open Access publishing is a different model, where the copyright remains with authors, who pay the journal to publish their articles which are then freely available. Launching this model in the UK, former science minister David Willetts argued it would boost the transparency of research institutions. Giving individuals, as well as industry, the “right-to-roam” academic journals would help people make better-informed choices (for example about their education) and could unleash the UK’s entrepreneurial spirit, he argued.

When open access was first introduced it initially had a reputation for vanity publishing – but as funding councils have embraced the idea it is becoming more mainstream. The UK funding agencies (Research Councils UK) have a policy that states that any outputs from research that it funds should be available via open access. Many other countries now also follow this model.

So, all the problems are resolved right? Well, no: There are concerns that institutions are still paying subscriptions and also are having to pay open-access charges.

Open access has a few variants. Gold open access is the model described above, where the paper is freely available on the journal’s website. There is also a Green option where you do not pay for open access but you are allowed to archive a version of your paper – typically the last version you submit before it is typeset – on your web site, or in an institutional repository, usually after some time. Institutions have to decide whether to adopt a Gold or a Green open access policy. The Romeo Sherpa is a very useful, enabling you to find out a journal’s position on open access.

Open access still struggles with its reputation. Only recently there was a report in the journal Science that: “Predatory publishers earned \$75 million last year”.

## The future

The internet and open access, combined with the publish-or-perish culture is changing the industry, arguably, faster than at any other time in history. What will it look like in ten years time?

I suspect that open access will be the norm, forcing universities to think about how to manage this and how they divert library funds from journal subscriptions to researchers to enable them to pay the open access charges. There is also the challenge of what to fund; all journals, only journals with an impact factor, or consider each discipline individually?

The contribution of the authors may also need to become more transparent, not only in reporting the use of publication consultants but also noting how each author has contributed. Perhaps it is a radical idea but the percentage contribution of each author could be given, which would also remove the problem of the order the authors.

The underpinning idea behind scientific publishing is peer review, in which research is forensically scrutinised by experts in the field before it’s published. But the process should also be transparent and fair. At the moment, there could be room for improvement.

Graham Kendall, Professor of Operations Research and Vice-Provost, University of Nottingham

## Should the contribution of a publication consultant be acknowledged?

This post was also posted in LinkedIn.

We have just had an article accepted in Science and Engineering Ethics which questions the need for publication consultants. Or, at least, if their services are used their contribution should be recognised either by being an author on the paper or by stating what their contribution was in the acknowledgments section.

The type of services these companies/individuals offer range from proof reading, conducting literature reviews, responding to reviewers comments, finding suitable journals and carrying out statistical analysis.

Our argument as to why these services are not needed, or should be acknowledged, is essentially three-fold.

1. If you are an experienced researcher (e.g. have been awarded a PhD) then you should be trained in carrying out all aspects of research and should not need to call on an outside agency.
2. Early career researchers should have the support of their institution. They are paying fees and/or are a registered student. In either case, they should be able to expect support from their supevisor, colleagues, Graduate School, mentors etc., rather than having to call on the services of a consultant, for which they have to pay.
1. If researchers have used a publication consultant, unless this has been acknowledged, then others reading the paper will not know that other people have contributed to the paper. This may be important for promotion committess, considering somebody for a job, when you are looking for collaborators etc.

We believe that if a publication consultant has been used this should be acknowledged on the paper, detailing what assistance they gave. Moreover, if the support provided would normally warrant authorship on the paper then the consultant should appear as an author.

Finally, we also suggest that publication consultants should be required to submit an annual return (perhaps just to the journal editor) stating which papers they have provided assistance for.

Once our paper is in press, we’ll provide the details.

## Why do researchers not read the aims and scope of a journal?

I also published this post on LinkedIn.

I am the editor-in-chief of a journal (not the journal you see here, that is a (very nice) public domain image from 1869) and, as such I see a lot of submissions. One of the biggest frustrations I face is when researchers submit papers that are simply not appropriate for the journal.

Sometimes their article title just has one of the same words as the title of the journal and, sometimes, the article title, keywords, abstract, content, references etc. are so obviously not appropriate to the journal that you wonder why they chose to submit to this journal?

I would love to know why researchers do this?

The paper will inevitably be rejected and it takes my time, the administator staff time, the Associate Editor’s time (the policy of the journal is that any reject must have the view of at least two people) and the researchers time. I do not see any benefit to any any of those people, and it simply means that the paper will not be published for even longer as authors have to wait for a reject decision.

The only good thing that comes out of it is that the journal I represent has a lower acceptence rate than it should do, which some people see as a sign of quality.

# Introduction

The 2015 UK General Election looks like being one of the closest, and hardest to predict, for many years. With 650 seats being contested, one party needs to win more than half the seats (326) to be able to form a government. Most, if not all, polls are predicting a hung parliament, with the likelihood that the UK will have another coalition government, though what form that will take is open to much debate.

It is not difficult to find predictions for the election result. They tend to fall into two categories; the percentage share of the vote or the number of seats that will be won by each party. Of most interest is the number of seats that will be won by each party, as this is what determines the formation of the next government.

# Wisdom of the Crowds

In 1907, Francis Galton reported in Nature an event that had taken place at a country fair, where around 800 people were asked to guess the weight of an ox. The average guess was 1,197 pounds. The actual weight was 1,198 pounds, which is close to the average guess to be considered just about spot on. Importantly, many of the people who participated could be considered experts, such as farmers and butchers, but many people were far from experts – just being people attending the fair. Also, importantly, not a single person guessed the correct weight and only one person guessed 1,197 and two people guessed 1,199.

This concept of the Wisdom of the Crowds was popularised in a 2004 book by James Surowiecki, arguing that the opinion of a large number of people will do better than the judgement of a few experts.

# 2010 General Election

Wisdom of the Crowds was used to predict the 2010 general election. Martin Boon, of ICM Research, showed that “that the Wisdom of Crowds approach at the 2010 general election would have produced the most accurate final pre-election prediction.

# Henretty and Jennings

Chris Henretty and Will Jennings have used the Wisdom of Crowds to predict the number of seats for each major party in the 2015 General Election. They surveyed 2,338 people, with 537 responding. They asked two questions, one about the percentage share of the votes and one about the number of seats for the major parties. Their report (published on 03 Mar 2015) gives the following predicted seats.

 Party Seats Con 278.4 Lab 282.3 Lib Dem 24.8 SNP 28.7 Greens 1.9 Plaid Cymru 3.3 UKIP 6.6 Others 13.4

# Our Data

Drawing inspiration from this study, we utilise other predictions, to see how it compares with the study of Henretty and Jennings. Our study looks at 24 different predictions, aggregating them to produce our predictions.

Our data is drawn from a variety of sources.

• The Henretty and Jennings study is used, recognising that it incoporates over 500 individual predictions.
• A recent BBC Panorama program asked Nate Silver for his predictions. Silver is an American statistician who has successfully predicted the outcome of the last two US presidential elections.
• Data was taken from several spread betting firms, taking the middle of the spread as their prediction.
• The London School of Economics asked a number of election forecasters at a conference they held on the 27 Mar 2015 for their predictions. These have been incorporated into our predictions below.
• Some newspapers publish predictions, and these are used in our model.
• Some on-line prediction web sites were used.

We decided against using the 2010 results, or the current parliamentary standings, although we show the predictions using these two additional pieces of data, just for completeness.

One issue that has to be considered is missing data. Predictors do not always provide predictions for all the parties, but provide an aggregated figure in Others for some of the parties. Some predictors also exclude Northern Ireland so they only supply 632 predictions, rather than the full 650. We work around this as best we can.

In order to calculate our predictions, we averaged all the polls under consideration. We normalise the figures for each party so that the total number of seats adds up to 650.

# Predictions

Our predictions are shown in the table below. The Excl. 2010 column shows the predictions when the 2010 results or the current parliamentary standings are not taken into account. The Incl. 2010 results are shown just for comparison.

 Excl. 2010 Incl. 2010 Con 279 281 Lab 274 272 Lib Dem 24 27 SNP 46 42 Greens 1 1 Plaid Cymru 3 3 UKIP 3 3 Others 20 21

The two sets of figures are reasonably close with the obvious differences being the higher prediction of the SNP and the lower prediction of the Lib Dems, which reflects the (potential) changing fortunes of the two parties since the last general election.

# Concluding Remarks

I guess, not surprisingly, we are also predicting a hund parliament, with the Conservatives having a slight lead over Labour. If our predictions are accurate, a coalition with the SNP would give a combined total of 325 seats – not quite the 326 needed to give an overall majority. Now that would be interesting!

I have been using Twitter for a few years now, but it has been a while since I blogged on this topic.

When I started tweeting I know that I would not be able to tweet every day, and weeks (or even months) could go by without me tweeting, and, to me, that would not give a very good impression.

Before I signed up I looked at how I could tweet automatically. You quickly come across services such a Hootsuite, which enable you to schedule tweets. This is a very nice service but you still have to do something on a regular basis. Like before, it is easy to forget, not have time etc. and you suddenly find yourself tweetless for weeks (or months).

That was the main reason I developed my own Twitter application. This enables me to do one tweet a day. This means that I am tweeting regularly and I hope that it is supplemented by a liberal smattering of ‘live’ tweets, as well as retweets.

I have automated two types of tweets.

1. I have a list of my publications, held as bibtex, and I tweet those.
2. I have another database, which holds (I hope) items of interest – let’s call them News.

Each day (it could be set up to do more), I choose (randomly) whether to tweet a News item, or a paper. Once that decision has been made, I just choose randomly from the relevant database/bibtex file.

But, this set me thinking. What would academics actually want from a Twitter service? I am specifically thinking about sending tweets, but I would also be happy to hear about reading them.

If you have any views, I would be really interested to know (as I might do it!).

# Computers are knocking on the door of the company boardroom

While women sitting on company boards remains a much-discussed topic, there is something new waiting to take a seat at the table: artificial intelligence, computers with company voting rights.

Deep Knowledge Ventures has appointed an algorithm called VITAL (Validating Investment Tool for Advancing Life Sciences) as a member of its board. It uses state-of-the-art analytics to assist in the process of making investment decisions in a given technology.

Of course, companies have used computer assisted analysis to analyse investment opportunities for a long time, but is the vision of a computer with equal voting rights as human board members a bit far-fetched?

## Defining artificial intelligence

What does the future hold with regard to the influence of computers on business decisions – and can they ever be used in place of a human board member? The Turing Test, formulated by Alan Turing in the 1950s, provides a strict interpretation of machine intelligence. A human participant must be unable to tell whether they are communicating (through a typed, text medium) with a computer or a human. If the human participant cannot reliably tell whether their conversation partner is a computer, then Turing would argue the computer has demonstrated intelligence.

Not everybody agrees that passing the Turing Test is enough for a computer to exhibit intelligence. In his Chinese Room argument, the Stanford philosopher John Searle described a closed room, into which a sentence written in Chinese is fed. A response emerges from the room, written in Chinese, that correctly answers the questions or conversational cues in the sentence submitted. The assumption could be made that inside the room is someone that can speak Chinese.

Instead, inside the room is a human who cannot speak Chinese but is equipped with manuals that exhaustively provide the appropriate Chinese characters to produce in response to those received. The argument holds that an appropriately programmed computer (the person in the room) could pass the Turing Test (by producing convincing Chinese) but would still not have an intelligent mind that we would regard as human intelligence (by understanding Chinese).

## A computer in the boardroom

If we want computers to make business decisions and even have equal voting rights on a company board, what would it have to do in order for the other board members to have confidence in its decisions?

Part of the challenge of the Turing Test is syntax versus semantics. Compare the sentences “Fruit flies like bananas” and “Time flies like an arrow”. The sentence structure is similar but the meaning is entirely different, making it a linguistic challenge.

Even a very simple conversation relies upon a substantial amount of linguistic knowledge and understanding. Consider the following questions:

• What was the result of the big match last night?
• I have K at my K1, and no other pieces. You have only K at K6 and R at R1. It is your move. What do you play? (these chess moves are from Turing’s original paper)
• What book do you think of if I say 42?

These might seem easy for humans to understand, but are challenging for a computer. Thankfully, a computer making business decisions is not faced with such a general task as the Turing Test. But if we are serious about having a computer as a full member of a company board, what are the hurdles that need to be addressed? Here is a (almost certainly not complete) list.

1. Access to LOTS of data: An automated approach to decision-making will require the use of big data. Company reports and accounts, economic data such as share prices, interest rates and exchange rates, and government statistics such as employment rates and house prices would all be obvious inputs. More subjective data such as newspapers, social media feeds and blogs might also be useful. Peer-reviewed scientific papers might also provide insight. Of course as always, the challenge with big data is to process the large quantities of data that will be be of different types (figures, text, charts), stored in different ways, and have missing elements.
2. Cost: Much of the data required is likely to generate significant costs. Social media feeds may be free (but not always), but stock market information, company accounts, government data, scientific papers and so on are generally commercial products that must be paid for. In addition, there is the cost of developing and maintaining the system. The algorithm is likely to require continual development by highly skilled analysts and programmers.
3. Complexity: Big data algorithms will be central to the boardroom decision support algorithm, but they will be underpinned by advanced analytics, many of which we are only just starting to understand and develop. To have a real impact there is likely to be some research required which would require staff with the relevant skills.

So, are we really at a point where a computer could take its place on the board? Technically it’s possible but the costs to develop and maintain, as well as subscribe to the data that is required, probably means that it is not within the reach of most companies and I suspect that the money would be better spent on a human decision maker – at least for now.

# The science that makes us spend more in supermarkets, and feel good while we do it

When you walk into a supermarket, you probably want to spend as little money as possible. The supermarket wants you to spend as much money as possible. Let battle commence.

As you enter the store your senses come under assault. You will often find that fresh produce (fruit, vegetables, flowers) is the first thing you see. The vibrant colours put you in a good mood, and the happier you are the more you are likely to spend.

Your sense of smell is also targeted. Freshly baked bread or roasting chickens reinforce how fresh the produce is and makes you feel hungry. You might even buy a chicken “to save you the bother of cooking one yourself”. Even your sense of hearing may come under attack. Music with a slow rhythm tends to make you move slower, meaning you spend more time in the store.

Supermarkets exploit human nature to increase their profits. Have you ever wondered why items are sold in packs of 225g, rather than 250g? Cynics might argue that this is to make it more difficult to compare prices as we are working with unfamiliar weights. Supermarkets also rely on you not really checking what you are buying. You might assume that buying in bulk is more economic. This is not always the case. Besides, given that almost half of our food is believed to be thrown away, your savings might end up in the bin anyway.

Strategies such as those above get reported in the media on a regular basis. Mark Armstrong analysed retail discounting strategies for The Conversation last year, for example, and the Daily Mail recently published a feature on making “rip offs look like bargains”.

You might think that awareness of these strategies would negate their effectiveness, but that doesn’t appear to be the case. It would be a strong person that does not give way to an impulse buy occasionally and, for the supermarkets, the profits keep flowing.

## Product placement

There are marketing strategies which you may not be aware of that also have an effect on our buying habits. Have you ever considered how supermarkets decide where to place items on the shelves and, more importantly, why they place them where they do?

When you see items on a supermarket shelf, you are actually looking at a planogram. A planogram is defined as a “diagram or model that indicates the placement of retail products on shelves in order to maximise sales”.

Within these planograms, one phrase commonly used is “eye level is buy level”, indicating that products positioned at eye level are likely to sell better. You may find that the more expensive options are at eye level or just below, while the store’s own brands are placed higher or lower on the shelves. Next time you are in a supermarket, just keep note of how many times you need to bend down, or stretch, to reach something you need. You might be surprised.

The “number of facings”, that is how many items of a product you can see, also has an effect on sales. The more visible a product, the higher the sales are likely to be. The location of goods in an aisle is also important. There is a school of thought that goods placed at the start of an aisle do not sell as well. A customer needs time to adjust to being in the aisle, so it takes a little time before they can decide what to buy.

You might think that designing a good planogram is about putting similar goods together; cereals, toiletries, baking goods and so on. However, supermarkets have found it makes sense to place some goods together even though they are not in the same category. Beer and crisps is an obvious example. If you are buying beer, crisps seem like a good idea, and convenience makes a purchase more likely. You may also find that they are the high quality brands, but “that’s okay, why not treat ourselves?”

This idea of placing complementary goods together is a difficult problem. Beer and crisps might seem an easy choice but this could have an effect on the overall sales of crisps, especially if the space given to crisps in other parts of the store is reduced. And what do you do with peanuts, have them near the beer as well?

Supermarkets will also want customers to buy more expensive products – a process known as “upselling”. If you want to persuade the customer to buy the more expensive brand of lager, how should you arrange the store? You still need to stock the cheaper options, for those that are really are on a budget. But for the customers that can afford it, you want them to choose the premium product. Getting that balance right is not easy. My colleagues and I are among the researchers striving to develop the perfect algorithm taking into account size, height and depth of shelves, to direct customers to the right product, at the right time.

Shoppers won’t always obey the science, but these techniques are retailers’ most effective tools in the fight for our weekly budget. The battle between supermarkets and their customers continues.

## What is your Erdös Number?

Paul Erdös (1913-1996) is one of the most prolific mathmeticians. He wrote over 1500 papers in his lifetime and collaborated with over 500 people. As a tribute, his friends created the Erdös number, which is a tongue in cheek way of asking how well you are associated with the top mathematicians.

Erdös himself has an Erdös number of 0. A co-author of Erdös has an Erdös number of 1, if you have written a paper with one of those co-authors you have an Erdös number of 2, and so on.

I found out recently that my Erdös is 3. That’s not too bad. I can never get an Erdös number of 1, but perhaps (one day) I might write a paper with somebody who has an Erd&ouml;s of 1, giving me anErdös number of 2.

If you are interested in finding out what your Erdös number is, I found two very good web sites:

1. The Erdös Number Project at http://www.oakland.edu/enp/

If you are interested in reading more about this fascinating man, and his life, I would highly recommend The Man Who Loved Only Numbers.

This post was originally published at the University of Nottingham.

# A day in the life of Pi

Most people have heard of the mathematical constant Pi (?), and will know that it’s roughly 3.14. Taking inspiration from these three digits, March 14 (3/14 in the US date format) is heralded as international Pi Day, first marked by US physicist Larry Shaw in 1988.

This year brings a unique opportunity to demonstrate an entirely unnecessary degree of zeal by marking Pi Day correct to nine decimal places on March 14, 2015, at 9.26am 53sec – corresponding to 3.141592653, the first 10 digits of Pi. If you’re too busy this weekend, you could book in July 22 – another way of expressing Pi approximately is the fraction 22/7.

Pi is calculated as the ratio between a circle’s circumference to its diameter.

Pi is always the same value, no matter the size of the circle, which makes it an important mathematical constant.

The ancient Babylonians calculated Pi as three by taking three times the square of the circle’s radius, later refining the value to 3.125. Archimedes of Syracuse (287-212 BCE) approximated Pi by inscribing polygons on the inside and outside of a circle. By increasing the number of sides of the polygons, Pi could be calculated to higher levels of accuracy.

Even today, calculating Pi to ever increasing levels of accuracy continues – it has now been found to an accuracy of over 13 trillion digits. There’s no reason to suspect this record will remain forever, even though only about 39 decimal places are sufficient for astronomical precision. There is no reason to be more precise for practical purposes but it is good scientific sport of sorts to strive to be ever more accurate.

## Some Properties of Pi

Pi is an irrational number, which means it cannot be accurately represented as a fraction, a/b, where a and b are integers. An approximation is to express it as 22/7 (3.1428…) which is inaccurate by 0.04025%. A closer approximation is 104348/33215, which has a far smaller error of 0.00000001056% but is still, technically, wrong.

Pi is also a transcendental number which, simplified, are numbers that cannot be reduced algebraically (more accurately a number that is not the root of any non-zero polynomial equation with rational coefficients).

The proof that Pi is transcendental was found in 1882, but it had been known for much longer that if Pi was transcendental then it would be impossible to square the circle – to construct a square with the same area as a circle.

## Putting Pi to use

Among the unusual uses for Pi is its relation to the nature of meandering rivers. A river’s path is described by its sinuosity, it’s tendency to wind from side to side as it traverses a plain. This is described mathematically as the length of its winding path divided by the length of the river as the crow flies. The average river has a sinuosity of about 3.14.

Albert Einstein actually made some observations about why rivers meandered as they did. He noticed that the water that flows faster around the outside of a bend, eroding that bank more quickly. This creates a larger bend. These bends eventually meet and the river forms a short cut through them. Hans-Henrik Stolum used these observations and noted a relationship with chaos theory, which suggests that, despite rivers straightening out as the rivers cut through the short cuts, the sinuosity tends to move back towards Pi.

Further examples of where Pi appears in the real world can be seen in a BBC item written for Pi Day 2008, and this New Scientist item written for the 2010 Pi Day. For example, Pi can be found in the measurements of the Great Pyramid of Giza, the angular distances of stars in the sky and in a song by Kate Bush. Included in the lyrics were the first hundred digits, or so, of Pi, but she went slightly wrong at around digit 50.

## How to celebrate Pi day?

If you fancy some Pi-related entertainment for Saturday, you could try:

• Looking up whether your birth date appears in the decimal places of Pi – mine does, starting at 200,703, although if you want to know my age you’ll have to look it up.
• Memorising the first digits of Pi. Piphilology, a system of mnemonics to help you remember the digits, may help. There are even piems (Pi poems) to help you remember. You may not beat the record-holder, which currently stands at 67,000 places.
• Examining the first million digits of Pi – you might see a pattern no one else has.
• Looking for Pi in everyday life. For example, it has featured in Mythbusters.
• Follow #piday2015 on Twitter, and see how others have marked the day in the past.

## … and finally

Albert Einstein, one of the greatest scientists the world has known, spent some time working on Pi as it related to rivers. Is it a coincidence that Albert Einstein was born on March 14, 1879? As he would have said himself, God doesn’t play dice.

For more slices of Pi, try a taste here or here.