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The Binomial test, sometimes referred to as the Binomial exact test, is a test used in sampling statistics to assess whether a proportion of a binary variable is equal to some hypothesized value. In this article, we explore the key features of this test and walk through an example test.

What are the hypotheses of the binomial test?

The hypotheses for the Binomial test are as follows:

The null hypothesis (H0) is that the population proportion of one outcome equals a specific hypothesized value (this can be denoted as π = πo).

The alternative hypothesis (H1) is that the population proportion of one outcome does not equal a specific hypothesized value (π ≠ πo).

Sometimes you may also want to test a null hypothesis of the population proportion as being greater than the hypothesized value specifically (or lesser than), rather than different in any direction, in which case you would perform a one tailed significance test, but more commonly the two tailed approach is used.

Note that there is no test statistic generated in a Binomial test as is common in other statistical tests such as the Mann-Whitney U Test or the unpaired Student's t-test, due to the p-value being calculated directly.

When to use the Binomial test

The Binomial test is used when a binary variable of interest (a variable that can take only two possible values e.g. mortality (dead/alive)) is being investigated and you have a hypothesized or expected value with which to compare it to. The test can only be used when sample size is small compared to the population about which you are trying to make an inference.

Changes to the shape of a Binomial distribution at varying values of the proportion of successes, p, and number of trials, n.

The Binomial test is derived from the Binomial distribution, which can be thought of as the distribution that is followed by the number of 'successes' or 'failures' in a certain number, n, of repeated independent experiments or 'trials'. In more statistical language, we can say that the distribution relies on the values of n and p (the probability of any trial being a success), and that these are the parameters of the Binomial distribution. It is useful to note that as the sample size (the value of n) increases, the distribution becomes more symmetrical and converges to a Normal distribution.

Binomial test assumptions

Assumptions for the Binomial test are as follows, and can be easily remembered using the 'BINS' acronym:

B - the variable of interest should be a binary outcome meaning it can take only one of two values (e.g. a coin toss (heads/tails), presence of a disease (yes/no), morality (dead/alive)). This is sometimes also referred to as a dichotomous variable.

I - observations should be independent, meaning that one observation should not have any bearing on the probability of another.

N - the experiment should have a fixed sample size denoted n.

S - all independent observations should have the same probability of having the outcome. This is similar to the independence assumption and can be achieved through random sampling.

Binomial test example

Suppose a population health researcher carries out a small random sample survey to estimate the prevalence (the proportion of a population affected) of herpes simplex virus (HSV), a common viral infection that causes genital and oral herpes. Members of the sample are selected at random with a total of 20 people selected ( n=20), are independent from one another and have the same probability of having the outcome, and with a binary outcome of interest (presence of HSV; yes/no).

The null hypothesis (H0) is that the proportion of survey participants (30%) with HSV is equal to 20% (0.2).

The alternative hypothesis (H1) is that the proportion of survey participants (30%) with HSV is not equal to 20% (0.2).

We can thus conceptualize this as a series of 20 independent trials with the proportion of people with the infection, p, following the Binomial distribution. Suppose in the survey it was found that 6 (30%) of the 20 participants had HSV. The probability of a given survey participant having the disease is therefore p=0.3. Suppose also that a previous survey found the prevalence of HSV to be 20% (this could be from the same population or a comparable population) - the researchers use this as the hypothesized value on which to run the Binomial test for the current survey proportion.

The next step is to run the Binomial test and generate a p-value, which denotes the probability of getting the proportion of people with HSV as extreme or more extreme than what was observed if the true p was equal to the hypothesized value. Statistical packages such as Stata, SPSS or R Studio can be relied upon to generate the Binomial test p-value, but for illustrative purposes the formula is detailed below. If we have n independent trials with probability of having HSV being p we can calculate the probability of the value being the hypothesized number of HSV cases, r (in this case r=4 as 20% of 20 is 4), using the following formula:

By plugging the values into the Binomial formula we get 0.196, the probability of 6 or fewer HSV cases out of 20 (one tailed test). Since our hypothesis of interest is whether the observed and hypothesized values differ in any direction, we would like to generate a two tailed test, and so we multiply by 2 to get a final p-value of 0.392.

The Binomial test formula features factorials represented by an exclamation point. These are calculated by multiplying the number by itself and then by every whole number through to 1. See here for a full hand calculation of the Binomial formula, and here for a convenient online calculator.

Using a significance level of α =0.05 we fail to reject the null hypothesis because p > 0.05 and conclude that there is no evidence of a statistically significant difference between the prevalence of HSV in the current survey compared with the previous survey given the sample size.

Elliot McClenaghan is a research fellow in Epidemiology and Medical Statistics at the London School of Hygiene & Tropical Medicine

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The Binomial test, sometimes referred to as the Binomial exact test, is a test used in sampling statistics to assess whether a proportion of a binary variable is equal to some hypothesized value. In this article, we explore the key features of this test and walk through an example test.

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How many people do you need to invite to a party to guarantee that at least 4 people know eachother?

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How many people do you need to invite to a party to guarantee that at least 4 people know eachother?

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If you invite six people to a party, either at least three of them will all know each other, or at least three will have never met before. But how many people do you have to invite to make sure that any given number of them are all friends, or all strangers?

This is a long-standing problem in combinatorics, the branch of mathematics that counts the ways in which a set of objects can be arranged. The optimal answer is surprisingly hard, and unknown except for a few simple cases. But a team of four mathematicians has now found the best upper limit so far - and the most substantial improvement to a previous solution dating back to 1935.

"All combinatorialists have tried hard to answer this question, and I think it's fair to say that it is one of the top two or three open problems in extremal combinatorics, or perhaps even the actual top one", tweeted mathematician Tim Gowers of the University of Cambridge, UK.

Gowers' remark followed a seminar in Cambridge on 16 March given by his departmental colleague Julian Sahasrabudhe. In a coordinated reveal, two of Sahasrabudhe's three coworkers - Robert Morris of the Institute of Pure and Applied Mathematics (IMPA) in Rio de Janeiro, Brazil, his doctoral student Marcelo Campos, and Simon Griffiths of the nearby Catholic Pontifical University (PUC) - also described their result at seminars at IMPA and in São Paulo, Brazil, on the same day. Otherwise, the work has so far been presented only in a 57-page preprint that has not been peer-reviewed.

Party people

The party problem was first stated in 1930 by the British mathematician and logician Frank Ramsey. One way to express it is to imagine a number R of people gathered together in a room, each of whom is either a friend or a stranger to the others. Say you are seeking a group of size k who are either all friends or all strangers. For a given k, what is the smallest R(k) that's guaranteed to satisfy that condition? It has long been known that any group of 6 people contains at least 3 friends or 3 mutual strangers, and that any group of 18 contains at least 4 friends or 4 strangers. But already for the number 5, the exact solution R(5) is unknown, except that it has to be between 43 and 48.

The problem can be mapped onto the mathematical theory of networks, the abstract objects made of nodes and links connecting them. Each node will represent a person at the party, and any two people are connected by a link, which is colour-coded - red if two people know each other, and blue if they do not. The question then becomes, how large does a network need to be to guarantee that it contains at least one all-blue or one all-red subset of a certain size?

Mathematicians Paul Erdős and George Szekeres showed in 1935 that R( k) is at most 4k. That bound is far from optimal: for example, for k = 4, it gives 4 ⁴ = 256, whereas it is known that 18 would suffice. But the formula at least gives an upper bound that's valid for any k.

The question that has tantalized combinatorialists since then is whether this upper bound can be any smaller - whether it can be shown that there is some bound C k for which C is less than 4. Even to lower it by a tiny amount, says Sahasrabudhe, "would be an exponential improvement over what was known [previously]", meaning that revising the bounds on C is the best possible way to improve on the earlier results.

The four researchers have now shown that the upper bound can be reduced at least to (3.9995)k. The result could have important ramifications for other areas of maths - in particular, for the study of networks that have an element of randomness, which can crop up in real-world problems ranging from epidemiology to optimization and scheduling problems, as well as for number theory and geometry.

Books and networks

The collaboration of the four researchers began in 2018, when Sahasrabudhe was visiting IMPA. Inspired by work on Ramsey's theorem by combinatorialist David Conlon of the California Institute of Technology in Pasadena, the team was able rather quickly to sketch an outline of what a proof might look like. "This turned out to be just the beginning of a long road ahead," says Sahasrabudhe.

Conlon's work suggested an approach involving auxiliary networks known as 'books'. The researchers set about devising an algorithm for constructing the books, but the challenge was to rule out 'pathological' exceptions to it. It wasn't until January of this year that they felt confident they had found a way around the problem. "By early February, we had a good idea of how we could use these ideas to obtain the exponential improvement", says Sahasrabudhe. "We were ready to go public in mid-March."

Their proof has not yet been checked by others, he cautions. "Now that the paper is public, the community will enter a period of validating, simplifying, digesting and improving our work. It will be exciting to see where others will take the ideas."

"The authors used elementary methods in an extremely clever way," Gowers told Nature. He expects that the result will stand up. "They had a huge incentive to be careful, and, I am told, were indeed extremely careful. All four are amazing mathematicians who have done wonderful work in the past." He adds that Sahasrabudhe's presentation had "a very convincing story about what was going on".

"The proof was very different from the kind of argument I had tried to find," Gowers adds, "and I don't think I would ever have found it."

"This is a problem that has proven persistently difficult, despite being enormously well studied for almost a hundred years," says Conlon. "Their progress is a stunning success."
doi: https://doi.org/10.1038/d41586-023-00840-5

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New maths formula answers long-standing party problem

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How many invitees guarantee at least a certain number all know each other? Formula is first major improvement since 1935. How many invitees guarantee at least a certain number all know each other? Formula is first major improvement since 1935.

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Mathematicians discover a shape that can tile a wall and never repeat

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The authors of a new preprint paper claim that they've discovered what's called an aperiodic monotile, a single shape that can cover a two-dimensional space with a pattern that never repeats itself exactly. One of the authors, Craig Kaplan, explains on Mastodon:

How small can a set of aperiodic tiles be? The first aperiodic set had over 20000 tiles. Subsequent research lowered that number, to sets of size 92, then 6, and then 2 in the form of the famous Penrose tiles.

Penrose's work dates back to 1974. Since then, others have constructed sets of size 2, but nobody could find an "einstein": a single shape that tiles the plane aperiodically. Could such a shape even exist?

Taylor and Socolar came close with their hexagonal tile. But that shape requires additional markings or modifications to tile aperiodically, which can't be encoded purely in its outline.

In a new paper, David Smith, Joseph Myers, Chaim Goodman-Strauss and I prove that a polykite that we call "the hat" is an aperiodic monotile, AKA an einstein. We finally got down to 1!

The full paper is here. You can play around with the tiles here & here and watch an animation of an infinite array of these monotiles.

If you're looking for a quick explanation of what aperiodic tiling is, check out the first 20 seconds of this video:

This video from Veritasium and this Numberphile one might also be helpful in understanding the concept. (thx, caroline)

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A Potential Major Discovery: An Aperiodic Monotile

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The authors of a new preprint paper claim that they've discovered what's called an aperiodic monotile, a single shape that can cover a two-dimensional space with a pattern that never repeats itself exactly. One of the authors, Craig Kaplan, explains on Mastodon: How small can a set of aperiodic tiles be?

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Qualifications © 2023 MME When you visit or interact with our sites, services or tools, we or our authorised service providers may use cookies for storing information to help provide you with a better, faster and safer experience and for marketing purposes.

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Jack Brown has uploaded a video for all Maths courses on all topics covered. A great place to start when your notes aren't making sense

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Got a TLMaths video request? Pop it in the SUGGESTION BOX

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Here is some advice from other students on how best to revise.
Copying notes and completing past papers is NOT the way to go!

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2019 will be the first year where a full cohort of students sits the new A Level Maths. Students studying the old modular spec had an almost endless supply of past papers that were an excellent revision resource. There are a limited number of past papers for the linear spec so planning your revision may not be as obvious. That's why we have produced this article that should hopefully provide you with useful guidance and resources.

( The best way to revise varies from person to person so while we hope this thread helps you, it will be up to you to make a revision plan that suits you.)

When to start?

This depends a lot on your confidence with the content as well as your priority of maths compared to your other A Levels. Remember that you need to revise everything you've learnt as part of a two year course across three very different areas (pure, mechanics & statistics) so make sure you don't leave it too late! If your teacher isn't planning to finish teaching the course until May or example, don't wait until they've finished before starting your revision of the content that you've already covered.

A month of revision may be enough but really you should aim for at least two/three months if you're targeting the top grades. If you can spend even a small amount of time every week going over the content from February/March then you will find full-time revision much easier when you start it later in the year. Again, this is just a guide and the earlier you can start your revision (without disrupting your learning of new topics) the better.

Revisit all of the topics

Before starting papers, you need to make sure that you are able to do the "standard questions" without too much thought. For example, you should be able to integrate a function using integration by parts or perform a hypothesis test using the normal distribution. Exam papers are not standard questons - you could be faced with a long modelling question in an exam and arrive at a function that requires integration by parts. You need to be confident enough with this topic to first notice that parts is required and be able to use the method.

Assuming that you have physical/online versions of textbooks designed for the new spec, go through them again revisiting questions and trying new questions that you didn't do the first time. Mixed/review exercises are useful for trying questions from different topics together. If you still don't have a textbook designed for the new spec then strongly consider buying one.

You can use online worksheets if you need more practice:

Solomon Worksheets : C1, C2, C3, C4 (these correspond to the new Pure syllabus with some differences)
Mathsmadeeasy
Resourceaholic

Also look through the exam board websites (not just your own) to find more A Level Maths resources:

If there are any topics that you feel you need more practice with then ask your teacher for more resources.

Understanding is key

For maths as opposed to a lot of other subjects, you should spend the majority of your revision time trying questions. However you must ensure that you understand everything in depth (this is especially important for mechanics and statistics). It is very common for students to be able to do a few textbook questions on a particular topic by following an example but then struggle when faced with a question that's a bit different because they lack the understanding. Make sure that this student isn't you! For example, in addition to being able to perform hypothesis tests involving the sample mean, could you explain to someone else what the sample mean probability distribution is and why its mean and variance make sense?

Make sure that you read through all of the explanations in your textbook carefully and ask your teacher if there is anything that you don't understand. Online videos are another great way to gain more understanding. ExamSolutions, MathsGenie and TLMaths have videos to explain every topic and are organised for the new spec. If you want written explanations then try DrFrostMaths and s-cool. Depending on how you like to revise, making notes can be a useful revision technique. Just make sure that you spend a lot more time doing questions than making notes.

Also consider buying a revision guide to help you consolidate all of your knowledge:

Find out what you need to know

For particular topics your teacher may have taught you more/less than what is required for the A Level Maths exam. That is why it's very important to look through your spec and see what you need to know and be able to do for the exam:

Also be clear on what will be contained in your formula book which will be given to you in the exams:

Just because a particular formula is in the formula book doesn't mean that you shouldn't learn it. For example, it is useful to know some of the trigonometry identities off-by-heart so that you can recognise when they can be used. Your teacher should be able to give you more guidance about which formulae are important to know and which others can be read from the formula book. When doing exam papers, make sure you have your formula book to hand.

Try loads of exam papers

One of the most important parts of A Level Maths revision is practicing as many exam papers as you can. After your final mock, if you haven't already been given all of the locked past/mock/practice papers for your exam board then speak to your teacher since they will have access and be able to share them with you. It could be that your teacher plans to give you them one at a time in the lead up to your exam which is also fine. The current lack of papers designed for the new spec means that it's very important that you try all of the official practice papers before the real exam.

For the new A Level, the content across the exam boards is the same which means that it's a good idea to try all of the exam board's unlocked papers. Be aware that the exam style could be quite different to your exam board plus the Large Data Set won't be the same, so it's best to ignore questions that refer to the LDS. There are specimen papers available online for other boards that you can try but unfortunately you probably won't be able to gain access to any of the locked papers.

Early on when trying papers, don't worry if you find them very hard and do worse than you expect. Exam papers are designed to be hard, especially for the new spec and you will improve as you get more confident with the content and develop exam technique. Make sure you try to do as many papers as you can under exam conditions without getting help and after finishing the paper, carefully read through the mark scheme and mark your papers accurately. It's very useful to know how mark schemes work and what examiners look for when marking papers. If there's something you got wrong then make sure you understand fully why you made the mistake so that you don't make a similar mistake again. It may be that you need to revisit the explanations of a topic to understand it better which is fine - the worst thing you can do is ignore something that you can't do.

Use old spec papers

Old modular past papers may not be as useful as they were before the spec change but they are still a great revision resource. It's a good idea to try some of these papers to find out how well you know the content and practice exam questions but be aware that the new linear exams are harder and there is a much bigger emphasis on modelling/problem solving, so don't rely solely on modular exams for your revision. If you decide to use these papers to practice individual topics then Physicsandmathstutor's Questions By Topic is the place to go. If you have any doubts about whether a question in an old modular exam could be in your exam then make sure you ask your teacher.

If you do decide to look at modular papers, here's a brief summary of papers to try:

For Pure try C1-2 papers (Year 1 topics) and C3-4 papers (Year 2 topics) and ignore topics that you haven't covered.

For Mechanics look at M1 papers as well as projectiles, non-constant acceleration and statics/moments questions in M2. M1/2 questions will prepare you well for the mechanics part of linear maths so try as many as you can.

For Statistics you may find that the modular exams are too different to be of any use but you might be able to pick out individual questions in S1/2 that are useful. Only look at S1/2 if you have run out of all other revision resources.

Edexcel have Modular to Linear and Linear to Modular mapping documents which provide a more in-depth summary. Be aware that the modular specs were not the same across exam boards so these documents only apply to Edexcel.

Still need more papers?

If you want more practice with new style papers then CrashMaths and MadAsMaths both have unofficial papers designed for the new spec for you to try. For Madasmaths, make sure you read all of the information to find out which papers correspond to A Level Maths. These papers are separated by difficulty so they can be useful for students who want more of a challenge.

Even with all of the specimen/practice papers etc. you may still find that you would like more practice with new style exam papers. Here are some links to practice papers that can be bought and you'll find others if you search online:

Don't forget about the large data set

The Large Data Set was designed to be used alongside the teaching of Statistics. Schools vary massively on how they use they make use of the LDS and there will be some students who are very familiar it by Year 13 and others who may have only seen it once or twice. The exam will assume the former and could ask questions that assume familiarity with terminology, context and limitations. Remember that you do not need to remember any of the data off-by-heart and any data required in an exam question will be provided to you.

You can revise statistics and the LDS at the same time by applying some of the methods that you have learnt to the data. For example, you could take a sample of the data and calculate summary statisics or find the PMCC. Your textbook will have many questions that refer to the LDS so make sure you try all of them and look at past/practice papers to get an idea of the types of questions that are asked. Here is a support document for the Edexcel LDS.

Ignore grade boundary predictions

When a new spec is introduced it can be very hard to predict grade boundaries. The 2018 boundaries were based on a small cohort of students plus this year's exams could be harder or easier so don't assume that your boundaries will correlate with the 2018 boundaries at all. The grades that you are given in school mocks are estimates so don't be disheartened if you are at a lower grade than you hoped - anything is still possible in the real exam. We know that not having marks to aim for is frustrating; just keep working hard to ensure that you achieve the highest mark that you can get.

Last but not least : use TSR!

If you've got a question about maths and your teacher is not around then try asking on this website. Use the A-levels forum to ask general questions about A Level Maths and use the Maths forum for academic help questions. There is also an A Level Maths Study Group, which is a dedicated thread for you to chat about A Level maths.

A Level Maths is not easy at the best of times but facing a brand new spec with few resources makes it an even bigger challenge. If you're ever struggling then remember that everyone else is in the same boat and the grade boundaries will reflect the fact that this spec is harder to revise for plus the exams are more challenging. Please share your struggles and successes with us and remember that we are here to support you if you ever need help. We wish you all the best of luck

(Please post comments and questions about this article here)

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2019 will be the first year where a full cohort of students sits the new A Level Maths. Students studying the old modular spec had an almost endless supply of past papers that were an excellent revision resource. There are a limited number of past papers for the linear spec so planning your revision may not be as obvious.

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Who: Year 12s considering studying maths at university
What: An opportunity to explore the subject with a small group of like-minded young people
Where: Winchester
Cost: £450
Application Deadline: end of April

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Winchester Maths Summer School returns in 2023 with stretching mathematical content and an exciting atmosphere, with unique opportunities for pupils to spend time with each other and the course leaders.

The residential course is open to Year 12 students from maintained sector schools who are considering studying maths at university. It provides an opportunity to explore the subject in a beautiful setting, with a small group of like-minded young people. The course will be led by specialist maths teachers from Winchester College, supported by undergraduate and graduate students from leading universities.

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Maths Summer School at Winchester College

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Maths Summer School at Winchester College is a course for Year 12 state school pupils who plan to study maths at university.

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An Imaginary Tale: The Story of i - Paul Nahin

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An Imaginary Tale: The Story of i - Paul Nahin

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For Double Mathematicians: Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.

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Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale , Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i . He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.

In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i . In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times.

Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.

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An Imaginary Tale: The Story of i [the square root of minus one]: The Story of √-1 (Princeton Science Library)

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Buy An Imaginary Tale: The Story of i [the square root of minus one]: The Story of √-1 (Princeton Science Library) With a New preface by the author by Nahin, Paul J. (ISBN: 9780691146003) from Amazon's Book Store. Everyday low prices and free delivery on eligible orders.

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If you are wanting to read just one book, make it this

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In 1997 I published my first book, a biography of a mathematical theorem. Below is a very brief summary of what it is about.

'I have discovered a truly marvellous proof,
which this margin is too narrow to contain...'

With these tantalising words the seventeenth-century French mathematician Pierre de Fermat threw down the gauntlet to future generations. Fermat's Last Theorem looked simple enough for a child to solve, yet the finest mathematical minds would be baffled by the search for the proof.

Over three hundred and fifty years were to pass before a mild-mannered Englishman finally cracked the mystery in 1995. Fermat by then was far more than a theorem. Whole lives had been devoted to the quest for a solution. There was Sophie Germain, who had to take on the identity of a man to conduct research in a field forbidden to females. The dashing Evariste Galois scribbled down the results of his research deep into the night before sauntering out to die in a duel. The Japanese genius Yutaka Taniyama killed himself in despair, while the German industrialist Paul Wolfskehl claimed Fermat had saved him from suicide.

Andrew Wiles had dreamed of proving Fermat ever since he first read about the theorem as a boy of ten in his local library. Whilst the hopes of others had been dashed, his dream was destined to come true - but only after years of toil and frustration, of exhilarating breakthrough and crashing disappointment. Fermat's Last Theorem is the true story of how mathematics' most challenging problem was made to yield its secrets is a thrilling tale of endurance, ingenuity and inspiration.

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In 1997 I published my first book, a biography of a mathematical theorem. Below is a very brief summary of what it is about. 'I have discovered a truly marvellous proof, which this margin is too narrow to contain...' With these tantalising words the seventeenth-century French mathematician Pierre de Fermat threw down the gauntlet to future generations.

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A far from average book: the real story behind the statistics on risk, chance and choice.

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Because chance and risk aren't just about numbers - it's about what we believe, who we trust and how we feel about the world around us. From a world expert in risk and the bestselling author of (and creator of BBC Radio 4's More or Less ), this is a commonsense (and wildly entertaining) guide to personal risk and decoding the statistics that represent it.

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The Norm Chronicles: Stories and numbers about danger

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Buy The Norm Chronicles: Stories and numbers about danger Main by Blastland, Michael, Spiegelhalter, David (ISBN: 9781846686214) from Amazon's Book Store. Everyday low prices and free delivery on eligible orders.

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Are you planning to study Maths or a Maths-related subject at university? You might be thinking about taking a Maths admissions test as part of your application.

Why do I need to take a Maths admissions test?

Mathematics is a competitive and demanding subject, and therefore universities use admissions tests as a way to ensure they have the right applicants on their course. Typically, tests are used for one of three reasons:

Shortlisting

Universities use them to shortlist applicants for interviews.

Offer reduction

Universities might give you a lower A Level offer if you do well in the test.

Final selection

Universities might require you to achieve a certain grade in a test (as well as A Levels) to meet your offer.

The tests

From looking at universities' websites or speaking to teachers and friends, you may already be aware of some of the Maths admissions tests. We're going to take a look at the following three:

STEP: Sixth Term Examination Paper

TMUA: Test of Mathematics for University Admission

MAT: Mathematics Admissions Test

It is important to check the details of the course(s) you're thinking of applying to, to see which ones are accepted or required.

STEP (Sixth Term Examination Paper)

STEP is used by the University of Cambridge, the University of Warwick and Imperial College London. Other universities sometimes ask candidates to take STEP.

There are also a number of candidates who take STEP as a challenge.

What's in the test and how is it scored?

There are two STEP papers; each takes 3 hours and is based on different A Level knowledge:

STEP 2: A Level Maths + AS Level Further Maths

STEP 3: A Level Maths + A Level Further Maths

The papers are graded on a 5-point scale. For more information, visit the STEP 'Scoring and results' web page.

When do I take STEP?

STEP papers are typically taken in June. See key dates and costs for STEP.

How do I prepare for STEP?

We provide free STEP preparation materials on our website, including past papers. We recommend starting your preparation by downloading the STEP specification as it provides a detailed overview of the test content.

To download all of our free materials and access others from the STEP Support Programme, visit the STEP preparation web page.

TMUA (Test of Mathematics for University Admission)

TMUA is used by 10 UK universities: University of Bath, University of Cambridge, Cardiff University, Durham University, University of Nottingham School of Mathematical Sciences, Lancaster University, University of Sheffield, London School of Economics and Political Science, University of Southampton and University of Warwick. Some of these universities make reduced offers to applicants who have done well on TMUA. Check their websites for details.

For most of the courses the test is not compulsory, however you're encouraged to take TMUA when applying to study Mathematics and Mathematics-related courses.

What's in the test and how is it scored?

TMUA is based on AS Level and Higher Level GCSE Mathematics. There are two TMUA papers; each takes 75 minutes and has 20 multiple-choice questions:

Paper 1: Questions assessing pure Maths in new unfamiliar situations

Paper 2: Questions assessing mathematical reasoning and logic

You are given a score based on your overall performance on both papers, which is given on a scale of 1.0 to 9.0. For more information, visit the TMUA 'Scoring and results' web page.

When do I take TMUA?

TMUA is typically taken at the end of October/beginning of November. See key dates and costs for TMUA.

Your results are sent directly to you. It is then up to you whether or not to share your results with universities. Therefore, it's risk-free - you have nothing to lose by taking TMUA because, if you're not happy with your result, your universities will never need to know.

How do I prepare for TMUA?

The test uses Mathematics you've already learned in your school studies, so there's no need to take a preparation course or to do much preparation beyond familiarising yourself with the test format and style of questions.

We provide the test specification, notes on Logic and Proof, and past papers with explained answers. To access all our free resources, visit the TMUA preparation web page.

MAT (Mathematics Admissions Test)

MAT is the University of Oxford's admissions test for undergraduate degree courses in Mathematics, Computer Science and their joint degrees. In conjunction with your UCAS application, Oxford uses MAT scores to shortlist applicants for interviews.

Imperial College London also uses MAT for all of its undergraduate Mathematics courses. Other universities also consider MAT.

What's in the test and how is it scored?

MAT is based on AS and A Level Mathematics and takes 2 hours 30 minutes.

Raw marks are given for MAT, which you can request from the universities as part of their feedback processes.

When do I take MAT?

MAT is typically taken at the end of October/beginning of November. See key dates and costs for MAT.

How do I prepare for MAT?

You will find everything you need to prepare for MAT on the universities' websites. Ensure you are familiar with the syllabus and test format by downloading specimen tests and past papers.

Which test should I take?

Ultimately, it depends on the universities you apply to.

Some universities require you to take a specific test, while others recommend tests that they accept, and therefore it's up to you which one you take. If you have the choice, you should consider which assessment will work best for you. However, you can take multiple tests if you want to.

Make sure you check the entry requirements of the course with the university you plan to apply to.

What if none of my universities mention a test?

If your chosen university doesn't require you to take a test and doesn't make reduced A Level offers, then there's no need to take any of these tests.

However, that doesn't mean you can't take one, and there are plenty of reasons why you should:

It allows you to demonstrate your aptitude and potential to study Maths and can therefore strengthen your university application, helping you to stand out from the crowd.

If you take TMUA, remember you don't have to share your results with your chosen universities unless you want to, so it's risk free!

It gives you the opportunity to experience what university Maths might be like.

It's a chance to challenge yourself!

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Maths admissions tests explained: STEP, TMUA and MAT

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Are you planning to study Maths or a Maths-related subject at university? You might be thinking about taking a Maths admissions test as part of your application. Mathematics is a competitive and demanding subject, and therefore universities use admissions tests as a way to ensure they have the right applicants on their course.

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Numerous problems, mathematical ideas and interesting mathematical history that you wouldn't otherwise have time to explore.

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So Much Maths, So Little Time. The CSF maths blog.

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People have used money – and made counterfeits - for thousands of years. Archimedes came up with a clever way of finding out if you’ve been cheated by a goldsmith. Making coins with the right proportions of the right metals presented a huge mathematical challenge for Fibonacci and other mathematicians in the middle ages.

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Professor Sarah Hart 11th October 2022
How can you detect a counterfeit coin? Why aren't coins square? Why is the wizarding money used in the Harry Potter universe so implausible? In this lecture we discuss the mathematics of coins. I'll focus on three main areas: creating, and testing for, the correct mix of metals, choosing the best coin shapes, and deciding on the ideal range of denominations.
Creating and detecting the real thing
Money has been used for thousands of years, and in that time there has been a huge range of materials in use, everything from gold and other precious metals to shells, leather, and of course paper. Many of these materials are "token" currency, in that they are not in themselves inherently of value. A five-pound note does not consist of five pounds worth of material. This is all very well if everyone agrees on the system. The advantage of gold and silver coins was that their value is intrinsic. If you have a gold coin then it is, literally, worth its weight in gold. Or at least, it is unless it is a counterfeit.

Counterfeiting is as old as trade, and one of the first stories about detecting it involves Archimedes. The story goes that the king (Hiero II of Syracuse, where Archimedes was from) had commissioned a golden crown and had given his goldsmith a certain amount of gold with which to make it. But Hiero suspected that the goldsmith may have in fact kept some of the gold for himself, making up the weight with silver. So, he asked Archimedes to find out the truth, without, of course, damaging the beautifully-wrought crown. An object made of a mixture of gold and silver, which is less dense than gold, would have a larger volume than one made of the same weight of gold. But it's surely impossible to measure the exact volume of so intricate an object as a crown. So how did Archimedes do it? Vitruvius, the Roman architect, wrote an account in his De Architectura (around 30 BCE) - it's a story we all remember: Archimedes sitting in his bath pondering the problem, suddenly realising that when he got into the bath he displaced exactly the same volume of water as the volume of his body, and then jumping out of the bath and running naked through the streets shouting "Eureka". The crucial realisation was that this is a way to measure volume without damaging an object (at least, not a metal one). Archimedes, goes the story, proceeded to take a block of pure gold the same weight as the crown, a block of pure silver the same weight as the crown, determine how much water they displace, and then compare how much water the crown displaced. With this information he could calculate exactly how much silver had been substituted. Legend has it that the goldsmith had indeed cheated the king.

The tale of Hiero's crown is an early instance of using the mathematics of density and volume to detect a forgery. The issue is somewhat simpler if you are comparing objects that are supposed to be the same size and composition, like coins. A suspect "gold" or "silver" coin can be weighed against one known to be genuine. It's also possible to determine the composition of a particular class of coin (say ones minted by a different country's mint) to determine exchange rates, by assaying. If you have a mixture of silver and gold plus other impurities you can determine the precise amounts by melting a sample, oxidising away the impurities, weighing what's left, which will be a mix of silver and gold, then dissolving the silver with nitric acid, to leave only the gold. You can then work out how much silver there had been, and what proportion of both of these were in the original alloy.

The other side of the coin, so to speak, is the challenge of the official mint: creating coins that have the precise legally required proportions of given metals. There's a nice paper by former Gresham geometry professor Norman Biggs on it if you are interested - the details are at the end of the transcript. An Italian mathematician and merchant called Leonardo of Pisa described techniques for producing alloys of a desired purity, or "fineness", in his extraordinary 13th century book Liber Abaci. You may not have heard of anyone called Leonardo of Pisa, but you probably have heard of him by his nickname Fibonacci. The so-called Fibonacci sequence was not invented by him, it had been known in India and elsewhere for a long time, but Liber Abaci was probably the first Western book to include it.
The shape of money
Now we know how to make a coin of the desired composition. But what does it look like? Most coins through history have been circular, though not all. There are examples of triangular and square coins, and other polygons with up to sixteen sides, as well as coins with round, square, and hexagonal holes. There are also rectangular and oval coins. Holes are useful for storing coins on strings or necklaces. Originally, coins would have been basically rounded lumps of metal. Then, when they began to be stamped with for example the Emperor's profile, they became flat. The round shape naturally forms a roughly circular shape when impressed with a stamp like this. A circular shape is more robust than one with lots of sharp corners, as these would wear down more quickly or possibly break off. A final reason is a more modern one. Vending machines find it easier to differentiate between different coins if each coin has a fixed diameter that's different from all the other coins. Circles, of course, almost by definition, are shapes with a constant width - their diameter at any point is exactly twice the radius. (This is useful for coins because of vending machines, but it also explains why most manhole covers are round. You don't want the cover to accidentally fall down the hole. With a circular cross-section, there is no way it can be aligned to be any narrower, so you are safe from that calamity.) But the circle is not the only shape with this property.

The same reasoning works for any higher odd number of sides. We can have a "pentagon of constant width", and also (pertinently for money) a "heptagon of constant width". This is exactly the shape of British 20p and 50p coins. (I have a soft spot for the 20p coin, which was introduced in 1982, because when I was a child it was the coin the tooth fairy brought me. My own children got one-pound coins: there's inflation for you.)

These curved shapes I've just described are called Reuleaux polygons (after the German engineer Franz Reuleaux who investigated their possible mechanical applications). The technique only works for odd numbers, because regular polygons with an even number of sides have the problem that every vertex is opposite another vertex, and the distance between two opposite vertices is always going to be greater than any other cross-sectional slice, however much you bulge out the sides. Reuleaux polygons do have some applications outside of coinage, too. For example, one property they have is that they can make a complete rotation inside a square such that at all times every edge of the square is in contact with the shape. A Reuleaux triangle drill bit will produce a very nearly square hole. This isn't just theoretical. One such drill bit was patented by Harry Watts in 1916 - it needs a special chuck to allow the centre of rotation of the bit to move, but with that in place, it covers over 98% of the square.

As well as the heptagonal coins in UK currency, there's also the Canadian one-dollar coin (nicknamed the "loonie" because it has a picture of a loon on one side): this is an eleven-sided Reuleaux polygon. Some Bermudian coins are Reuleaux Triangles. There's the 1997 one dollar coin, as well as several "Bermuda Triangle" commemorative coins.

This is cute, but why do it? Why not just make circular coins - it's easier after all? Of course, it's just fun to have a bit of variety. It's also harder to counterfeit a Reuleaux polygon than a simple circle, but there's one possible other reason, and it's to do with areas.

Firstly, there's a theorem called Barbier's Theorem that says all shapes of constant width s have the same perimeter, namely πs . This is fairly easy to see for circles (diameter s means circumference πs ) and Reuleaux polygons, whose perimeters are arcs making up half the circumference of a circle of radius s , but far from clear that it holds in general. However, it does. On the other hand, the areas of these shapes can vary. A Reuleaux triangle with constant width s can be thought of as the superimposition of three 1/6th slices of a circle of radius s . So you can find the area by adding three of these together, but then remembering that this counts the equilateral triangle inside three times. So the formula for the area is

This is less than any other Reuleaux polygon and also less than a circle of diameter s , which would have area 1 4 π s 2≈0.79 s 2 . In fact it can be shown that the Reuleaux triangle has the smallest area of any shape of constant width s . So, in terms of material used, for a given width you save money with the Reuleaux triangle. You might like to work out the area of a Reuleaux pentagon with constant width s . Is it better or worse than a circle?

Of course, anyone familiar with British coins will have spotted that not all coins can possibly have constant width, because we have a twelve-sided coin: the £1 coin. This has an even number of sides, so it can't (and indeed doesn't) have constant width. So, what's going on? It's not the first twelve-sided coin in British currency. Many years ago the threepenny bit had twelve sides. However, you'll notice that the modern £1 coin does have slightly curved sides. This makes the discrepancy between its smallest (23.03mm) and largest (23.43mm) widths small enough for vending machines to cope. The old threepenny coin would have caused more issues (if you wish, you can calculate the difference between smallest and largest width to confirm this), and that is why the curved sides were adopted.

So, the coin is a "near enough" design that satisfies vending machines. That doesn't mean I have to like it.
Coin values and denominations
Now we have coins made of the right alloys and the right shape. Our final currency decision is what denominations they should have.

Essentially everyone has decimal currency nowadays - in other words, a currency where units are subdivided into powers of ten sub-units. For example, Euros, and US dollars, are divided into 100 cents; pounds sterling are divided into 100 pence. The Nigerian Naira is 100 Kobo, one Russian rouble is 100 kopeks (and has been for over 300 years). And so on.

There are only a handful of exceptions, mostly where the subdivisions are now too small to be used, or where the currency is not in use for other reasons. For example the scudo is still the official currency of the Sovereign Military Order of Malta, though everyone uses Euros and has done for many years. The scudo is divided into 12 tari, with 1 tari equivalent to 20 grani.

But historically, the situation was reversed - almost every currency was non-decimal. Why? And why did it change? Just as an example, if we look at historical British money, we had shillings divided into twelve old pennies, and twenty shillings in the pound, or sovereign. (And 21 shillings in a guinea.) Pennies in turn were divided into four farthings (from the word "fourthling"). They were legal tender until 1960. So, you can divide up a shilling into two, three, four, six whole numbers of pennies, and into greater subdivisions with halfpennies and farthings. A few hundred years ago there were coins worth ¼ penny (the farthing), ½ penny, 1 penny, 2 pence (the half groat), threepence, fourpence (groat), sixpence, a shilling, half a crown (2 shillings and sixpence), a crown (five shillings), and gold sovereigns (1 pound). This is simplifying matters slightly because the values of the silver and gold coins fluctuated with the prices of these metals - that's why guineas ended up being worth slightly more than a pound - and the purity of the coins; silver coins had ever-increasing amounts of copper, with the currency being repeatedly debased because of it. The values weren't formally fixed until later.

You could divide a pound equally, using whole numbers of pennies, into 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120 and 240. There's not as much flexibility in the decimal pound, which can be divided equally into 1, 2, 4, 5, 10, 20, 25, 50, and 100. That's the big advantage of pre-decimal shillings and pounds. The big disadvantage is the arithmetic. Once you start writing down calculations, it is very cumbersome to use our decimal number system for calculations in a system that has twelve pence to the shilling and twenty shillings to the pound. (Consider also our baffling imperial units of sixteen ounces to a pound and fourteen pounds to a stone, but obviously (!) twenty fluid ounces to a pint in the UK, and sixteen in the US. Our American friends have a rhyme for converting water volume to weights: "a pint's a pound the world around". Only it isn't because in England "a pint of water weighs a pound and a quarter".)

If you think pre-decimal currency is bad, spare a thought for Harry Potter and his Hogwarts friends. In J.K. Rowling's books, wizard money consists of golden galleons, silver sickles and bronze knuts. There are 17 sickles in a galleon, and 29 knuts in a sickle. This is an insane system that wouldn't even allow amounts like half a galleon, never mind the formidable arithmetic that would be required to make transactions.

Back in the real world, people have been making the case for decimalisation for hundreds of years. In 1696, Christopher Wren, former Gresham Professor of Astronomy, proposed the introduction of a silver "noble", divided into 10 "primes" and 100 "seconds", "which Centessimal Division will be very proper for Accounts". The objections to this idea and others were mostly that, desirable though it would be, people would never be able to learn a new system. Of course, if a country is brand new that's less of a problem. The United States dollar, introduced in 1792, was divided not into "pieces of eight" like the Spanish silver dollar from which it took its name and initial value, but from the start into tenths (dimes) and hundredths (cents). The debate in Britain rumbled on, and in the 19th century several commissions and a lot of parliamentary time was taken up with it. The florin (two shillings) was introduced in 1849 as a possible precursor to decimalisation, because it's a tenth of a pound (and some of them - the so-called Gothic Florins - even had "one tenth of a pound" written on them). There was even a double florin, briefly, which is equivalent to our modern 20 pence piece, because it is a fifth of a pound. The double florin, though, worth four shillings, was very close in size to, and easily confused with, the five-shilling crown, so it was withdrawn. It was only minted for four years, from 1887-1890 [15]. After yet more Royal Commissions, including in 1918 and 1961, decimalisation was finally agreed, and the big switchover happened on 15th February 1971.

So, what coins should we have? It makes sense for whole numbers of smaller coins to go into the pound. This means the possibilities for coins are 1, 2, 4, 5, 10, 20, 25, 50. In the UK we have six of these eight: 1p, 2p, 5p, 10p, 20p, 50p (as well as £1 and £2 coins). A 4p coin would not be very useful - it can already be made with two 2p coins, and is too close in value to the 5p. In the US, they have fewer coins: 1 cent, 5 cents (nickel, though at 75% copper and 25% nickel, perhaps it should be called a copper), 10 cents (dime), and 25 cents (quarter). There is a dollar coin but, due to the enduring popularity of the dollar bill, it hasn't caught on.

Is one system better than the other? The best systems combine simplicity (a small number of different denominations) while allowing change to be given using a small number of coins. The simplest system - just have 1p coins - would require us all to lug great big bags of coins around all the time. At the other extreme, if there were a coin of every value we would need gigantic cash registers for all the different coins and would no doubt never have a 47p coin when we needed one. As a measure of how efficient different systems are, Dr Adam Townsend at the University of Durham suggested using the average number of coins required to make up each amount of change up to the point where the currency switches to notes. For the US, that's one dollar. For the UK, it's five pounds. We can think about whether this is the right measure or not. Some currencies have notes starting at much lower values than others, and this may skew the results. I modified the Python script he recommended to calculate not just the mean but the median number of coins, and I also felt that a US - UK comparison would be more informative if we calculated the change up to the same amount in both currencies: 99p/99¢. My code, and links to his articles, are at the end of this transcript.

For values of change from 1p up to and including 99p, the current UK system needs a (mean) average of 3.43 coins, and a median of 3 coins. For up to 99¢ the US system needs a mean of 4.75 coins and a median of 5. So it's much worse, with some amounts needing eight or even (for 94¢ and 99¢) nine coins (the UK system never needs more than 6 coins). Adam Townsend calculates average change up to and including the smallest bill so it calculates 4.75 for the US system but 4.61 for the UK.

By the way, if we had eight fingers like most cartoon characters, rather than ten, we'd probably use a base 8 system with 8×8 pennies in the pound. Then we could have 1, 2, 4, 8, 16 and 32p coins, and with a £1 =64p and £2 = 128p coin, we'd need only a mean (and median) of 4 coins to make change up to the smallest note being a £4 = 256p note, and you'd need at most one of each coin for any given amount of change. But I digress!

Should the US switch to 20 cents rather than quarters? Then the coins are 1, 5, 10, 20. The mean average change is 5.05 coins, so the US should stick with quarters, unless it wants to add other coins, like a 2 cent or 50 cent coin. Does this mean the should UK switch to "quarters" rather than 20p? Curiously, this would make absolutely no difference to the amount of change received. The outcome is identical, given the other coins we have.

All this assumes we are getting change in the most efficient way. Anyone who has used an automated checkout at a supermarket knows that's not always the case. Because self-checkout machines are often based on global designs, and most countries have fewer different coins than us, these machines can take any of our eight coins but can usually only give six different kinds of coins in change. Most often, this is 1p, 2p, 5p, 20p, £1, £2. If we just look at amounts of change up to 99p, we now find we get 4.75 coins on average (or a median of 5), which is much more than we would get from a human. If we test amounts up to £4.99, we get a mean of 5.91 and a median of 6. Interestingly, as Adam Townsend points out, things would be much better if the coins instead were 1p, 2p, 5p, 20p, 50p, £2. For amounts up to 99p you would get a mean of 3.84 coins and a median of 4.

There are some issues with coins. The 5p is irritatingly small. The 1p and 2p are irritatingly heavy. (If you are outside the UK, you probably have similar gripes with your own currency.) Is it time to ditch some of them? We used to have halfpenny coins, even post-decimalisation, and we even had farthings within living memory. The farthing was discontinued in 1960 and the halfpenny in 1984. Now, I've done some calculations, with the aid of the Bank of England's inflation calculator (at time of writing, as it only covers complete years, the calculator only compares a given year to 2021 prices.) It turns out that £1 in 1960 would be worth £16.18 today (or actually at the end of 2021). One farthing is a quarter of an old penny, and so 1/960th of a pound. So that farthing would be worth 1.7 new pence as at 2021, and with inflation at over 10% in 2022, that's only going up. By that token we should get rid of the 1p coin and we are close to the point where we might consider getting rid of coppers completely.

How about halfpenny coins? They were discontinued in 1984, and £100 in 1984 is worth £263.02 today. So a 1984 halfpenny would be worth 1.3p today. Again, a strong case for getting rid of the 1p coin. (In fact, halfpennies would have been phased out sooner, except that the Bank of England were worried that it might worsen inflation, as all the prices might be rounded up to the nearest penny.) The halfpenny coin was not missed as a coin in itself, but some had other uses for it. A letter to the Times begged "let not its existence be imperilled. It is indispensable for levelling off pendulum clocks". We would manage perfectly well without the penny. The Canadian mint ceased production of the Canadian cent (fondly known as the penny) in 2013. They didn't have a two cent coin so their smallest coin is now 5 cents. Items can still be priced in pennies, and electronic transactions can still be done to the nearest penny, but change is given to the nearest five cents.

One day we may not use coins at all. But I think that day is still (happily) quite far off. I hope you've enjoyed this mathematical guide to coins. In my next lecture on "maths and money" we look at how maths can help us to find the best strategies when we buy, sell, bargain, barter, and bid at auctions. I hope to see you then.

© Professor Sarah Hart 2022
Further Reading

Hiero's Crown: for a description of how the account of Vitruvius could have been correct, see The Vitruvius Tale of Archimedes and the Golden Crown, by Amelia Carolina Sparavigna, https://arxiv.org/ftp/arxiv/papers/1108/1108.2204.pdf. Professor Chris Rorres at Drexel University has posted the relevant passages of Vitruvius, the treatise of Galileo, and other sources, alongside English translations, at https://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html

For a much more in-depth history than I can give on alligation and linear programming, read the excellent paper Linear programming from Fibonacci to Farkas, by Norman Biggs, which you can download at http://eprints.lse.ac.uk/106596/.

If you want to read Liber Abaci and your Latin is on the rusty side, there's an English translation (we only had to wait 800 years!) Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation, by Laurence Sigler (Springer New York; 2002, ISBN 978-0387407371).

Marianne Freiberger wrote a good article in 2017 about the introduction of the 12-sided £1 coin in Plus magazine: https://plus.maths.org/content/new-1-coin-gets-even

You can read a short summary of the history of decimalization at the Royal Mint website: https://www.royalmint.com/discover/decimalisation/decimal-debate/

Adam Townsend's entertaining blogs about coin denominations in different currencies and about vending machine change appeared in Chalkdust magazine.

https://chalkdustmagazine.com/blog/forget-a-new-1-pound-coin-we-need-a-1-pound-23-coin/

https://chalkdustmagazine.com/blog/self-service-machines-give-awful-change/
He also made a programme on change for Radio 4's "Boring talks" about it, which is well worth a listen: Boring Talks Episode 33: Change https://www.bbc.co.uk/programmes/p06xvmpz

The Python script I used is adapted from the one Adam Townsend used. My script is below, and you can play around with it at one of the many free online Python compilers, for example https://www.w3schools.com/python/trypython.asp?filename=demo_compiler

coins = [1,2,5,20, 100, 200]

rangecoin=99

min_coin = [100] * (rangecoin+1)

min_coin[0] = 0

for min_of_i in range(rangecoin+1):

for c in coins:

if c <= min_of_i and (min_coin[min_of_i - c] + 1 < min_coin[min_of_i]):

min_coin[min_of_i] = min_coin[min_of_i - c] + 1

print(min_coin)

print(sum(min_coin)/rangecoin)

import statistics

print(statistics.median(min_coin))

The Bank of England's inflation calculator https://www.bankofengland.co.uk/monetary-policy/inflation/inflation-calculator can show how prices compare to today, for any year since 1209!

Image Credits

Unless otherwise specified, the technical diagrams are drawn by me and are ©Sarah Hart; photos/images are either taken by me or are public domain images.

Hiero II coin, photo by Sailko, CC BY 3.0, via Wikimedia Commons. https://en.wikipedia.org/wiki/Hiero_II_of_Syracuse

Reuleaux triangle drill bit gif by LEMeZza, CC BY 3.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Rotation_of_Reuleaux_triangle.gif

Images of coins (20p, 50p, £1, Canadian "loonie" dollar, Bermuda currency), are copyright the various national mints, but their use falls under the fair use policy.

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People have used money - and made counterfeits - for thousands of years. Archimedes came up with a clever way of finding out if you've been cheated by a goldsmith. Making coins with the right proportions of the right metals presented a huge mathematical challenge for Fibonacci and other mathematicians in the middle ages.

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The e16plus Newsletter is aimed at 16 – 19 year olds who are interested in mathematics. It aims to keep readers up to date with mathematical news, people, and activities. It features careers that make use of mathematics, mathematics in use in the real world, mathematical history, competitions, and much more.

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The e16plus Newsletter is aimed at 16 - 19 year olds who are interested in mathematics. It aims to keep readers up to date with mathematical news, people, and activities. It features careers that make use of mathematics, mathematics in use in the real world, mathematical history, competitions, and much more.

# data[].attributes.attachment_link.description
The e16plus Newsletter is aimed at 16 - 19 year olds who are interested in mathematics. It aims to keep readers up to date with mathematical news, people, and activities. It features careers that make use of mathematics, mathematics in use in the real world, mathematical history, competitions, and much more.

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A small, easy-to-read book about the wonders of Maths

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1089 and All That: A Journey into Mathematics

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David Acheson's extraordinary little book makes mathematics accessible to everyone. From very simple beginnings he takes us on a thrilling journey to some deep mathematical ideas. On the way, via Kepler and Newton, he explains what calculus really means, gives a brief history of pi, and even take...

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preparing for the test of mathematics for university admission

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test of mathematics for university admission preparation

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Download specimen papers to help you prepare for the Test of Mathematics for University Admission (TMUA).

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We offer this each year in early October so look out for emails or ask your teacher.
This link will also provide you with links to past papers to help you prepare

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The Senior Mathematical Challenge is a 90-minute, multiple-choice Challenge.

It encourages mathematical reasoning, precision of thought, and fluency in using basic mathematical techniques to solve interesting problems.

The problems on the Senior Mathematical Challenge are designed to make students think. Most are accessible, yet still challenge those with more experience.

Date: Tuesday 4th October 2022 Please note: Schools on China Standard Time (UTC +08:00) need to sit the Challenge on Wednesday 5th October 2022

Eligible year groups: England, Wales and Overseas - Year 13 or below, Scotland S6 or below, Northern Ireland - Year 14 or below

Time allowed: 90 minutes

Entry fee: £13 for a pack of 10 entries
Entries open now!
Paper posted entry deadline (having papers posted out): Midnight, Wednesday 14th September 2022

Paper download and online entry deadline: You can now place paper and online entry orders right up to the day of the Challenge, including on the day itself. You just need to make sure you leave yourself enough time to pre-populate, download and print your answer sheets and question papers.

How to take part

1. Entries must be made by registered UKMT centres on behalf of students.

2. You can order a paper or online version or a combination of both (schools outside of the UK can only participate on paper by downloading and printing their question papers)

3. The Challenge must be sat under exam conditions.

4. We mark both the paper and online entries electronically and notify you of the results.

Purchasing entries

Entries into the Senior Mathematical Challenge can be purchased via your online Challenges Profile.

If you purchase online entries you will be provided with an online login for each student, which enables them to log in to the Challenges site and participate in an online format.

UK Schools - If you purchase paper entries we will send you question papers via post, however answer sheets will need to be downloaded and printed in school. If you miss this deadline, you will still be able to purchase paper entries however you will need to download and print your question papers alongside your answer sheets. Your student's answer sheets will then need to be uploaded to your Challenges Profile for us to mark electronically. Please note that we are no longer able to mark hard copy answer sheets, so please ensure these are not sent back to the UKMT via post.
Overseas Schools - If you purchase paper entries you will need to download and print both your question papers and answer sheets. Your student's answer sheets will then need to be uploaded to your Challenges Profile for us to mark electronically. Please note that we are no longer able to mark hard copy answer sheets, so please ensure these are not sent back to the UKMT via post.

Access the Challenge Handbook to help navigate you through hosting the Senior Mathematical Challenge.

Preparing for the Challenge

We hope that taking part in the Senior Mathematical Challenge is a stimulating and rewarding experience for all students, whether or not they have prepared.

You can utilise our previous past papers, solutions and investigations here. They are a great resource containing high-quality problems designed to encourage mathematical thinking and prepare students for the Challenge. We are also delighted to be working with DrFrostMaths to help teachers use past problems in the classroom.

We also sell downloadable past papers and books which can help with your preparation.

Scoring

Scoring rules for the Senior Mathematical Challenge are as follows:

All participants start with 25 points

4 marks are awarded for each correct answer

1 mark is deducted for each incorrect answer

Answers left blank will be marked as 0

Awards and follow-on rounds

All students will receive a certificate of participation. To recognize the highest performers in the Challenge, we award the top-scoring 66% of participants Bronze, Silver and Gold certificates in the ratio 3:2:1. Students that are 'Best in Year' or 'Best in School' will also receive a badge to denote this on their certificates. Certificates have the student names pre-populated and can be downloaded and printed as soon as results are released.

In addition, we invite around 1,000 of the very highest performers to take part in the British Mathematical Olympiad Round 1 and around 6,000 to take part in the Senior Kangaroo.
The 2021 award thresholds were:
85+ Gold, 66+ Silver, 49+ Bronze. Thresholds for invitation to Senior Kangaroo 85+ and BMO1 108+ depending on eligibility.

Contributing to our work

As a charity, we do not make a profit from entry fees. We spend all the money we raise on advancing the education of young people in mathematics. If you would like to support us further then please consider making a donation.

Our volunteers are at the heart of everything UKMT does from creating the Challenges to running our Enrichment activities. If you would like to find out more about volunteering with us please do get in touch.

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Promoting a love of problem solving The Senior Mathematical Challenge is a 90-minute, multiple-choice Challenge. It encourages mathematical reasoning, precision of thought, and fluency in using basic mathematical techniques to solve interesting problems. The problems on the Senior Mathematical Challenge are designed to make students think.

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The world turns on symmetry -- from the spin of subatomic particles to the dizzying beauty of an arabesque. But there's more to it than meets the eye. Here, Oxford mathematician Marcus du Sautoy offers a glimpse of the invisible numbers that marry all symmetrical objects.

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Marcus du Sautoy: Symmetry, reality's riddle

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The world turns on symmetry -- from the spin of subatomic particles to the dizzying beauty of an arabesque. But there's more to it than meets the eye. Here, Oxford mathematician Marcus du Sautoy offers a glimpse of the invisible numbers that marry all symmetrical objects.

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The latest breaking news, comment and features from The Independent.

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Sharing maths and science in the news

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Quanta Magazine (@quantamag) * Instagram photos and videos

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172K Followers, 342 Following, 1,776 Posts - See Instagram photos and videos from Quanta Magazine (@quantamag)

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Thinking about Mathematics - Stewart Shapiro

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Thinking about Mathematics - Stewart Shapiro

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Stewart Shapiro admirably provides an accessible introduction to contemporary thinking in mathematics, while avoiding caricature of the technicalities. His ease with the subject and lucid style makes this book a succinct introduction to a fascinating intellectual discipline

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Select Your Cookie Preferences

We use cookies and similar tools that are necessary to enable you to make purchases, to enhance your shopping experiences and to provide our services, as detailed in our Cookie Notice. We also use these cookies to understand how customers use our services (for example, by measuring site visits) so we can make improvements.

If you agree, we'll also use cookies to complement your shopping experience across the Amazon stores as described in our Cookie Notice. This includes using first- and third-party cookies, which store or access standard device information such as a unique identifier. Third parties use cookies for their purposes of displaying and measuring personalised ads, generating audience insights, and developing and improving products. Click 'Customise Cookies' to decline these cookies, make more detailed choices, or learn more. You can change your choices at any time by visiting Cookie Preferences, as described in the Cookie Notice. To learn more about how and for what purposes Amazon uses personal information (such as Amazon Store order history), please visit our Privacy Notice.

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Thinking About Mathematics: The Philosophy of Mathematics

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This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an his...

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preparing for mathematics admissions test

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Links to specimen papers and past papers with answers, to help prepare for the Mathematics Admissions Test (MAT).

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The undergraduate mathematics major offers a foundation for students interested in understanding how to approach problems with mathematical solutions. The mathematical economics major offers students a program that combines mathematics, statistics and economics.
Where Can Mathematics Take Me? A Career in Numbers? You Do the Math
Where Can Mathematical Economics Take Me? What skills does studying mathematics develop?
Mathematics is challenging, rewarding and fun. It is both logical and creative. Students who major in mathematics have a variety of opportunities. The mathematics major prepares students for traditional pursuits such as graduate study, teaching and work as an actuary. Students with a love of math find a mathematics major can be combined with a pre-professional curriculum or a major in the sciences or engineering to provide a strong background for graduate study or employment in a field related to mathematics.
Career Opportunities
The mathematical economics major provides an opportunity for students with an interest in mathematics and business or economics to combine these interests. The Department of Mathematics offers both a Bachelor of Science and a Bachelor of Arts degree. Each degree can be taken with Option A: Mathematics or Option B: Mathematical Sciences. The mathematics option is chosen by most students. The mathematical sciences option combines the study of mathematics, statistics and computer science and prepares students for careers involving the applications of mathematics. A mathematics minor is available for students who would like to continue their study of mathematics, while majoring in another field. In addition to the bachelor's degrees, the Department of Mathematics offers programs leading to the Master of Arts (M.A.), Master of Science (M.S.), and the Doctor of Philosophy (Ph.D) degrees.

critical thinking

problem solving

analytical thinking

quantitative reasoning

ability to manipulate precise and intricate ideas

construct logical arguments and expose illogical arguments

communication

time management

teamwork

independence

For more information about careers, go to the following:
Career opportunities are unlimited for mathematics majors. They may pursue graduate education, career paths in business, science or technical fields or disciplines such as social services, education and government. Some of the occupations that mathematics majors enter include:
A Compilation of Competence: The Mathematical Economics Degree

accountant

actuary

computer programmer

doctor

engineer

investment manager

lawyer

government research and laboratories

theoretical mathematician

mathematician

numerical analyst

statistician

teacher

market researcher

systems analyst

banking

government

space/aircraft industry

What skills does studying mathematical economics develop?
For more information about career opportunities, contact career counselor Jamie Johnson at jjjohn4@email.uky.edu or (859) 257-4023.
Career opportunities
The mathematical economics major offers students a degree program that combines mathematics, statistics and economics. In many ways, the mathematical economics program parallels the engineering philosophy. It combines the quantitative methods of mathematics with an applied science in order to solve real problems. With the ever-increasing importance of the service sector in our economy, the mathematical economics degree will prove to be a valuable asset. The program will give the student an opportunity to study a fascinating collection of ideas and it will also provide the student with very marketable skills.

capability in mathematics and statistics

aptitude in problem solving skills

ability to communicate with public speaking and written assignments

capacity to collaborate with others

For more information about careers, go to the following:
In today's increasingly complicated international business world, a strong preparation in the fundamentals of both economics and mathematics is crucial to success. This degree program is designed to prepare a student to go directly into the business world with skills that are in high demand, or to go on to graduate study in economics or finance. A degree in mathematical economics would prepare a student for the beginning of a career in operations research or actuarial science. Other occupations include but are not limited to the following:

economist

management accountant

actuary

bank examiner

market research analyst

financial analyst

marketing/sales manager

financial planner

claims examiner

investment manager

international trade specialist

real estate investor

statistician

teacher

professor

For more information about career opportunities, contact the UK Career Center.

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The undergraduate mathematics major offers a foundation for students interested in understanding how to approach problems with mathematical solutions. The mathematical economics major offers students a program that combines mathematics, statistics and economics. Where Can Mathematics Take Me? A Career in Numbers? You Do the Math Where Can Mathematical Economics Take Me?

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A group challenge to to January with registration in October

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Ritangle is a competition designed for teams of students aged 16‌-‌18 who are currently studying either:

A level Mathematics

the International Baccalaureate

Scottish Highers

qualifications with equivalent content.

No knowledge of maths beyond A level Mathematics is expected.
Register your team and view questions

Students who take part are so engaged and excited by it, every year. They live and breathe it for the duration. And they learn so much from being involved in it.

Registration

from 3 October

Teams can register for this year's competition

Stage 1 questions

3 October - 5 December

For the first four weeks, one question is released every Monday. For the following 5 weeks, three questions will be released on Mondays, Wednesdays and Fridays. The final question of Stage 1 will be posted on Monday 5 December.

Every correct answer reveals a piece of information that helps solve the Final Task.

Final Task

6 December, 16:00 (GMT)

Final challenges released.

Use the information you've gained from solving the previous questions to reveal new challenges and solve the Final Task.

Deadline: Final Task

13 December, 16:00 (GMT)

Deadline for submitting answers to the Final Task

Approximating Roots of Equations

General advice about Microsoft Excel

A spreadsheet, which can be programmed with Visual Basic

General advice about GeoGebra

Graph plotter, spreadsheet, general mathematical calculator, Computer Algebra System, programmable

General advice about using the Casio fx-CG50

General advice about Python

Programming language

General advice about Maplesoft suite of desktop, online and mobile CAS tools

Getting started with MATLAB

Download a classroom poster

Promote the competition to your students

Ritangle is a free maths competition aimed at teams of sixth form/ college students running from early October to mid-December. The 'Stage 1' questions are released one at a time across this period; the answers unlock information to solve the final multi-part question.

Stage 1 questions can be viewed on the Integral Twitter feed and registered teams can access them through their team login.

The final question is available only to registered teams through their team login.

Yes!

One student from the team must register the team. The team should agree who this will be to ensure there is only one entry per team. This student is the Team Captain. Every student can only be in one team, including the Team Captain. Once registration opens, you'll be able to do so on the Ritangle webpage and a team must be registered in order to submit answers.

School/college name

Team Captain's name

Team Captain's e-mail address

Nominating Teacher's name

Nominating Teacher's e-mail address

No, each student can only be in one team, including the Team Captain.

Rules will be published at the start of the competition.

One of the most important rules is about not sharing answers. Please don't share your answers outside your team, online or anywhere else; others are having fun finding them!

Please note: anyone found leaking the answers or clues inappropriately and anyone found making intentional use of such leaks will be disqualified from the competition.

Yes! We encourage the use of technology; spreadsheets, graphing software and programming can be particularly useful. Any use of online resources/search engines is also permitted.

When we explore maths for ourselves these days, we may have a spreadsheet, a graphing program, a CAS program and a programming environment open simultaneously, and we want Ritangle to have that flavour. We think it is reasonable to expect students to be able to use a graphing program and a spreadsheet. Programming knowledge will not, however, be required for the final question.

On the other hand, we accept that when working against the clock (such as while working on the final task) methods using technology may have outpaced more traditional methods. We bear this in mind when constructing tasks.

Please note: any team repeatedly attempting to use an automated process to systematically enter answers until the correct answer is found will be disqualified from the competition.

This is tricky. We have tried 9:00 (2016-2018 inclusive) but this favours students whose schools are happy for them to go off timetable for part of the day. On the other hand, 16:00 makes life difficult for other students, who may be travelling at that time. Support for each of these start times has been about equal amongst participants.

Our solution is to make the final task harder to complete quickly, so that the winning time lies between two days and seven days. This is a delicate balancing act.

We have considered dropping the race-to-the-line element of the final task, and having a more qualitative task for those scoring fully on the final question. We have decided to shelve this alternative for now.

Yes! Judges may request to see evidence of working out. Answers to questions are also required at a later stage in the competition. We recommend that each team nominates a member as Team Archivist to keep the team's working for the questions as they are attempted.

Every team that submits a creditable answer to the final question will receive a certificate, emailed to their Nominated Teacher.

The winning team will receive a maths hamper, containing a mix of prizes for individual team members and their school. It includes a free year-long subscription to Integral, and a trophy for the school or college of the winning team. We reserve the right to substitute a prize of equivalent value but of less weight/bulk should the winning team be based abroad

Please note: the following is subject to the team providing sufficient evidence (in the opinion of the judges) of their working.

Answers for the final question will be accepted up until the date published in the Information and Rules document.

Scenario 1

The team scoring most highly on the final question will be declared the winner, subject to their working satisfying the judges.

Scenario 2

If there is more than one team with the highest score on the final question, the winner will be chosen, from such teams, as the team that submitted those answers first, subject to their working satisfying the judges.

Scenario 3

In the event that the above does not determine a winner, the winning team will be chosen at the discretion of the judges based on the answers submitted.

If there is a possibility a team has won, the judges will contact the Nominating Teacher to verify details of the team and their participation. If the answer is satisfactory to the judges then that team will be the winner. The decision of the judges is final. No discussion will be entered into regarding the decision over who is the winner. The winning team and the answers will be announced on this webpage and on the Integral Twitter feed.

Please contact ritangle@mei.org.uk

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Ritangle is a competition designed for teams of students aged 16‌-‌18 who are currently studying either: A level Mathematics the International Baccalaureate Scottish Highers qualifications with equivalent content. No knowledge of maths beyond A level Mathematics is expected. Register your team and view questions Students who take part are so engaged and excited by it, every year.

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Why I fell in love with monster prime numbers

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Why I fell in love with monster prime numbers

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They're millions of digits long, and it takes an army of mathematicians and machines to hunt them down -- what's not to love about monster primes? Adam Spencer, comedian and lifelong math geek, shares his passion for these odd numbers, and for the mysterious magic of math.

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Adam Spencer: Why I fell in love with monster prime numbers

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They're millions of digits long, and it takes an army of mathematicians and machines to hunt them down -- what's not to love about monster primes? Adam Spencer, comedian and lifelong math geek, shares his passion for these odd numbers, and for the mysterious magic of math.

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We humans can be a scary acquaintance for whales in the wild. This includes marine biologists tagging them with measuring devices to understand them better. These experiences can make whales behave ...

Children do better at maths when music is a key part of their lessons, an analysis of almost 50 years of research on the topic has ...

Researchers created a generic framework that enables an engineer or scientist to evaluate the effectiveness of defense schemes that seek to limit a hacker's ability to learn secret information ...

New research from Dartmouth takes a fresh look at game theory to show that being uncooperative can help people on the weaker side of a power dynamic achieve a more equal outcome -- and even inflict ...

Children who receive high-quality child care as babies, toddlers and preschoolers do better in science, technology, engineering and math through high school, and that link is stronger among children ...

Researchers developed a machine-learning technique that can autonomously drive a car or fly a plane through a very difficult 'stabilize-avoid' scenario, in which the vehicle must stabilize ...

Researchers are trying to make the process more efficient and definitive by using artificial intelligence, combined with a classic mathematical method for quantifying web-like networks, to determine ...

Researchers explore the role of uneven coffee extraction using a simple mathematical model. They split the coffee into two regions to examine whether uneven flow does in fact make weaker espresso. ...

Researchers wanted to understand precisely how blackworms execute tangling and ultrafast untangling movements for a myriad of biological functions. They researched the topology of the tangles. Their ...

ChatGPT faced off against students on accounting assessments. Students scored an overall average of 76.7%, compared to ChatGPT's score of 47.4%. On a 11.3% of questions, ChatGPT scored higher ...

In the quantum world particles can instantaneously know about each other's state, even when separated by large distances. This is known as nonlocality. Now, A research group has produced some ...

A researcher has solved a nearly 60-year-old game theory dilemma called the wall pursuit game, with implications for better reasoning about autonomous systems such as driver-less ...

Mathematical research brings new understanding of crowd formation and ...

A new study finds that high school students identify more with math if they see their math teacher treating everyone in the class equitably, especially in racially diverse schools. While the ...

Researchers test cognitive abilities of the language model ...

Octopus arms coordinate nearly infinite degrees of freedom to perform complex movements such as reaching, grasping, fetching, crawling, and swimming. How these animals achieve such a wide range of ...

Theoretical chemists have developed a theory that can predict the threshold at which quantum dynamics switches from 'orderly' to 'random,' as shown through research using ...

Using artificial intelligence (AI), researchers can now follow cell movement across time and space. The method could be very helpful for developing more effective cancer ...

A new study finds that a 200-year-old technique called Fourier analysis can reveal crucial information about how the form of artificial intelligence called deep neural networks (DNN) learn to perform ...

If scaled up successfully, the team's new system could help answer questions about certain kinds of superconductors and other unusual states of ...

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math; mathematics; math-oriented; mathematical model; mathematical theory; mathematical method; maths; mathematician; mathematician's; math model; math education; classroom; geometry; geometric shape; statistics; trigonometry

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Explore a wide range of recent research in mathematics. From mathematical modeling to why some people have difficulty learning math, read all the math-related news here.

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The stars of the show include mathematicians and other guests from around the world. Topics range from the sublime to the ridiculous… from historic discoveries to latest breakthroughs.

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Listen to this episode from The Numberphile Podcast on Spotify. A breakthrough paper is published about using artificial intelligence to do mathematics - we discuss it with DeepMind's Alex Davies and Professor Marcus du Sautoy.

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Daily geometry puzzles

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Brilliant.org (@brilliantorg) * Instagram photos and videos

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437K Followers, 63 Following, 290 Posts - See Instagram photos and videos from Brilliant.org (@brilliantorg)

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Simon Singh offers the first sweeping history of encryption, tracing its evolution and revealing the dramatic effects codes have had on wars, nations, and individual lives. From Mary, Queen of Scots, trapped by her own code, to the Navajo Code Talkers who helped the Allies win World War II, to the incredible (and incredibly simple) logisitical breakthrough that made Internet commerce secure, The Code Book tells the story of the most powerful intellectual weapon ever known: secrecy.

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When I wrote my first book, Fermat's Last Theorem, I made a passing reference to the mathematics of cryptography. Although I did not know it at the time, this was the start of a major interest in the history and science of codes and code breaking, which has resulted in a 400-page book on the subject, an adaptation of the book for teenagers, a 5-part TV series, numerous talks and lectures, the purchase of an Enigma cipher machine and the development of an interactive crypto CD-ROM.

You can find information about TV series and the free CD-ROM version of the book (and lots more) in the Cryptography section of the site. In this section, however, you will find details about my book on cryptography (The Code Book).

I have not included any material about the teenage version of The Code Book, but you can find it at Amazon.co.uk and Amazon.com.

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When I wrote my first book, Fermat's Last Theorem, I made a passing reference to the mathematics of cryptography.

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STEP past papers are available to download by year as zipped files. Each zipped file contains all papers. Answers are included from 2004 onwards, and the Examiner's Report also from 2007.

Please note, some past papers are based on the old test specifications. The style of the questions will remain unaltered, and most of the STEP questions from previous papers can be used for preparation.

New specifications are available in the STEP resources section below for 2019 onwards, and students are advised to consult these when preparing.

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Download free past papers, mark schemes and other resources to help you prepare for STEP (Sixth Term Examination Paper) Mathematics.

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© Copyright AGCAS & Graduate Prospects Ltd

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What can I do with a mathematics degree? | Prospects.ac.uk

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Discover what jobs, internships and courses you can do with a degree in mathematics.

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Start date: Early October

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Skip to main content

The School of Mathematical Sciences presents the 19th National Cipher Challenge

In the lead up to WWII the British intelligence services were flooded with reports of Nazi spies and sympathisers. With war fast approaching and limited resources the task of vetting all these reports was passed to a fledgling security organisation known as BOSS. As a new recruit you are given the job of decoding the secret messages that will expose the enemy agents. It is a race against time and your country needs you ...

Login / register Go to Challenges

Latest News

THIS WAS THE FIRST BOSS PUZZLE THIS YEAR WITH A BOSS RATING OF 4. Maybe you found it easier than Harry, and if so, hat tip to you! Either way, we hope you enjoyed it! In the completed puzzle the numbers 0 through 9 appear in each of the top six rows of the grid so that they give the column sums at the bottom and no number appears in twice in adjacent cells, including diagonal adjacency.

Read on

BOSS PUZZLE CHALLENGE 8 - open until Sunday 13th Dec 9am

ANOTHER GREAT GRID PUZZLE FROM OUR GUEST PUZZLEMASTER THE LETTER WRIGGLER, whose identity, like those of all our officers and agents, is a closely guarded secret. TLW first became interested in ciphers after reading Simon Singh's The Code Book in 1998. He found the now NCC web page way back in the early 2000's and has enjoyed every years session since, but until 2019 he remained a 'lurker' never registering (because it was for schools only and did not think [...]

Read on

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Skip to main content The School of Mathematical Sciences presents the 19th National Cipher Challenge In the lead up to WWII the British intelligence services were flooded with reports of Nazi spies and sympathisers. With war fast approaching and limited resources the task of vetting all these reports was passed to a fledgling security organisation known as BOSS.

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Perfect if you also study Spanish, Eduardo Saenz de Cabezon answers a question that teachers dread to hear: What is math for?

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With humor and charm, mathematician Eduardo Sáenz de Cabezón answers a question that’s wracked the brains of bored students the world over: What is math for? He shows the beauty of math as the backbone of science — and shows that theorems, not diamonds, are forever. In Spanish, with English subtitles.

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Mathematics News - Math News, Mathematical Sciences

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Phys.org provides the latest news on mathematics, math, math science, mathematical science and math technology.

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Tim Harford and the More or Less team try to make sense of the statistics which surround us.

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Listen to this episode from More or Less: Behind the Stats on Spotify. Daily advances in the technology of artificial intelligence may leave humans playing catch-up - but in at least one area we can still retain an edge, mathematics.