Category: mathematics

Does the thought of p-values and regressions make you break out in a cold sweat? Never fear – read on for answers to some of those burning statistical questions that keep you up 87.9% of the night.

• What are my hypotheses?

There are two types of hypothesis you need to get your head around: null and alternative. The null hypothesis always states the status quo: there is no difference between two populations, there is no effect of adding fertiliser, there is no relationship between weather and growth rates.

Basically, nothing interesting is happening. Generally, scientists conduct an experiment seeking to disprove the null hypothesis. We build up evidence, through data collection, against the null, and if the evidence is sufficient we can say with a degree of probability that the null hypothesis is not true.

We then accept the alternative hypothesis. This hypothesis states the opposite of the null: there is a difference, there is an effect, there is a relationship.

• What’s so special about 5%?

One of the most common numbers you stumble across in statistics is alpha = 0.05 (or in some fields 0.01 or 0.10). Alpha denotes the fixed significance level for a given hypothesis test. Before starting any statistical analyses, along with stating hypotheses, you choose a significance level you’re testing at.

This states the threshold at which you are prepared to accept the possibility of a Type I Error – otherwise known as a false positive – rejecting a null hypothesis that is actually true.

• Type what error?

Most often we are concerned primarily with reducing the chance of a Type I Error over its counterpart (Type II Error – accepting a false null hypothesis). It all depends on what the impact of either error will be.

Take a pharmaceutical company testing a new drug; if the drug actually doesn’t work (a true null hypothesis) then rejecting this null and asserting that the drug does work could have huge repercussions – particularly if patients are given this drug over one that actually does work. The pharmaceutical company would be concerned primarily with reducing the likelihood of a Type I Error.

Sometimes, a Type II Error could be more important. Environmental testing is one such example; if the effect of toxins on water quality is examined, and in truth the null hypothesis is false (that is, the presence of toxins does affect water quality) a Type II Error would mean accepting a false null hypothesis, and concluding there is no effect of toxins.

The down-stream issues could be dire, if toxin levels are allowed to remain high and there is some health effect on people using that water.

Do you know the difference between continuous and categorical variables?

• What is a p-value, really?

Because p-values are thrown about in science like confetti, it’s important to understand what they do and don’t mean. A p-value expresses the probability of getting a given result from a hypothesis test, or a more extreme result, if the null hypothesis were true.

Given we are trying to reject the null hypothesis, what this tells us is the odds of getting our experimental data if the null hypothesis is correct. If the odds are sufficiently low we feel confident in rejecting the null and accepting the alternative hypothesis.

What is sufficiently low? As mentioned above, the typical fixed significance level is 0.05. So if the probability portrayed by the p-value is less than 5% you reject the null hypothesis. But a fixed significance level can be deceiving: if 5% is significant, why is 6% not?

It pays to remember that such probabilities are continuous, and any given significance level is arbitrary. In other words, don’t throw your data away simply because you get a p-value of 6-10%.

• How much replication do I have?

This is probably the biggest issue when it comes to experimental design, in which the focus is on ensuring the right type of data, in large enough quantities, is available to answer given questions as clearly and efficiently as possible.

Pseudoreplication refers to the over-inflation of degrees of freedom (a mathematical restriction put in place when we calculate a parameter – e.g. a mean – from a sample). How would this work in practice?

Say you’re researching cholesterol levels by taking blood from 20 male participants.

Each male is tested twice, giving 40 test results. But the level of replication is not 40, it’s actually only 20 – a requisite for replication is that each replicate is independent of all others. In this case, two blood tests from the same person are intricately linked.

If you were to analyse the data with a sample size of 40, you would be committing the sin of pseudoreplication: inflating your degrees of freedom (which incidentally helps to create a significant test result). Thus, if you start an experiment understanding the concept of independent replication, you can avoid this pitfall.

• How do I know what analysis to do?

There is a key piece of prior knowledge that will help you determine how to analyse your data. What kind of variable are you dealing with? There are two most common types of variable:

1) Continuous variables. These can take any value. Were you to you measure the time until a reaction was complete, the results might be 30 seconds, two minutes and 13 seconds, or three minutes and 50 seconds.

2) Categorical variables. These fit into – you guessed it – categories. For instance, you might have three different field sites, or four brands of fertiliser. All continuous variables can be converted into categorical variables.

With the above example we could categorise the results into less than one minute, one to three minutes, and greater than three minutes. Categorical variables cannot be converted back to continuous variables, so it’s generally best to record data as “continuous” where possible to give yourself more options for analysis.

Deciding which to use between the two main types of analysis is easy once you know what variables you have:

ANOVA (Analysis of Variance) is used to compare a categorical variable with a continuous variable – for instance, fertiliser treatment versus plant growth in centimetres.

Linear Regression is used when comparing two continuous variables – for instance, time versus growth in centimetres.

Though there are many analysis tools available, ANOVA and linear regression will get you a long way in looking at your data. So if you can start by working out what variables you have, it’s an easy second step to choose the relevant analysis.

Ok, so perhaps that’s not everything you need to know about statistics, but it’s a start. Go forth and analyse!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Sarah-Jane O’Connor*

 

The names of the trigonometric ratios tangent and secant are derived from the Latin “to touch” and “to cut” – the tangent to a figure is a line that touches it in one place, where a secant cuts through it in two or more. But how are these geometric terms related to the ratios that bear their names? The answer can be shown using the diagram at the top of the post – a diagram that used to be a standard one in high school trig text books.

Consider the acute angle BAC. Allow |AC| = 1, and construct a unit circle about A that goes through C. Construct a tangent to this circle at C, and extend the segment AB so that it meets this tangent at E. So, the segment CE lies on the tangent while the segment AE lies on the secant of the unit circle formed around BAC. ACE is a new right triangle that contains the original BAC.

The tangent of BAC is BC/AB (opposite/adjacent), but if we now look at the second triangle ACE, we see tht it is also given by (CE/AC)=(CE/1)=CE – the tangent is measured by the segment of the tangent, CE. Similarly, the secant of BAC is given by AC/AB (hypoteneuse/adjacent), but again turning to the second triangle ACE, we see that this is (AC/AB)=(AE/AC)=(AE/1)=AE – and the secant is provided by the length of the secant, AE.

This treatment was taken from the book “Plane Trigonometry and Tables” by G. Wentworth, published in 1903. In some of the texts of this era, the “primary” trigonometric ratios were sin, sec, and tan (rather than sin, cos, and tan), perhaps owing their primacy to constructions like the one described above.

The cosine was considered a secondary trigonometric ratio – its name coming from the phrase “complement’s sine.” Along with the usual ratios, texts often presented several convienience ratios that are now antiquated, such as the versedsine vrsin(x) = 1-cos(x) and the half-versed sine or haversine hvrsn(x)= (1/2)vrsin(x).

The most fundamental trigonometric ratio has the most obscure name. It is generally claimed that the word “sine” comes from Latin word for “bend,” but some have suggested that the word is ultimately derived from the name of the curve formed by the gathering of a toga, or from the Latin word for “bowstring.” In Arithmetic, Algebra, Analysis, Felix Klein states that “sine” represents a Latin mis-translation of an Arabic word, but does not go on to explain its origins any further.

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*Credit for article given to dan.mackinnon*

Converting hundreds of compositions by Johann Sebastian Bach into mathematical networks reveals that they store lots of information and convey it very effectively.

Johann Sebastian Bach is considered one of the great composers of Western classical music. Now, researchers are trying to figure out why – by analysing his music with information theory.

Suman Kulkarni at the University of Pennsylvania and her colleagues wanted to understand how the ability to recall or anticipate a piece of music relates to its structure. They chose to analyse Bach’s opus because he produced an enormous number of pieces with many different structures, including religious hymns called chorales and fast-paced, virtuosic toccatas.

First, the researchers translated each composition into an information network by representing each note as a node and each transition between notes as an edge, connecting them. Using this network, they compared the quantity of information in each composition. Toccatas, which were meant to entertain and surprise, contained more information than chorales, which were composed for more meditative settings like churches.

Kulkarni and her colleagues also used information networks to compare Bach’s music with listeners’ perception of it. They started with an existing computer model based on experiments in which participants reacted to a sequence of images on a screen. The researchers then measured how surprising an element of the sequence was. They adapted information networks based on this model to the music, with the links between each node representing how probable a listener thought it would be for two connected notes to play successively – or how surprised they would be if that happened. Because humans do not learn information perfectly, networks showing people’s presumed note changes for a composition rarely line up exactly with the network based directly on that composition. Researchers can then quantify that mismatch.

In this case, the mismatch was low, suggesting Bach’s pieces convey information rather effectively. However, Kulkarni hopes to fine-tune the computer model of human perception to better match real brain scans of people listening to the music.

“There is a missing link in neuroscience between complicated structures like music and how our brains respond to it, beyond just knowing the frequencies [of sounds]. This work could provide some nice inroads into that,” says Randy McIntosh at Simon Fraser University in Canada. However, there are many more factors that affect how someone perceives music – for example, how long a person listens to a piece and whether or not they have musical training. These still need to be accounted for, he says.

Information theory also has yet to reveal whether Bach’s composition style was exceptional compared with other types of music. McIntosh says his past work found some general similarities between musicians as different from Bach as the rock guitarist Eddie Van Halen, but more detailed analyses are needed.

“I would love to perform the same analysis for different composers and non-Western music,” says Kulkarni.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Karmela Padavic-Callaghan*

Relating mathematics to questions that are relevant to many students can help address its image problem, argues Eugenia Cheng.

Mathematics has an image problem: far too many people are put off it and conclude that the subject just isn’t for them. There are many issues, including the curriculum, standardised tests and constraints placed on teachers. But one of the biggest problems is how maths is presented, as cold and dry.

Attempts at “real-life” applications are often detached from our daily lives, such as arithmetic problems involving a ludicrous number of watermelons or using a differential equation to calculate how long a hypothetical cup of coffee will take to cool.

I have a different approach, which is to relate abstract maths to questions of politics and social justice. I have taught fairly maths-phobic art students in this way for the past seven years and have seen their attitudes transformed. They now believe maths is relevant to them and can genuinely help them in their everyday lives.

At a basic level, maths is founded on logic, so when I am teaching the principles of logic, I use examples from current events rather than the old-fashioned, detached type of problem. Instead of studying the logic of a statement like “all dogs have four legs”, I might discuss the (also erroneous) statement “all immigrants are illegal”.

But I do this with specific mathematical structures, too. For example, I teach a type of structure called an ordered set, which is a set of objects subject to an order relation such as “is less than”. We then study functions that map members of one ordered set to members of another, and ask which functions are “order-preserving”. A typical example might be the function that takes an ordinary number and maps it to the number obtained from multiplying by 2. We would then say that if x < y then also 2x < 2y, so the function is order-preserving. By contrast the function that squares numbers isn’t order-preserving because, for example, -2 < -1, but (-2)² > (-1)². If we work through those squaring operations, we get 4 and 1.

However, rather than sticking to this type of dry mathematical example, I introduce ones about issues like privilege and wealth. If we think of one ordered set with people ordered by privilege, we can make a function to another set where the people are now ordered by wealth instead. What does it mean for that to be order-preserving, and do we expect it to be so? Which is to say, if someone is more privileged than someone else, are they automatically more wealthy? We can also ask about hours worked and income: if someone works more hours, do they necessarily earn more? The answer there is clearly no, but then we go on to discuss whether we think this function should be order-preserving or not, and why.

My approach is contentious because, traditionally, maths is supposed to be neutral and apolitical. I have been criticised by people who think my approach will be off-putting to those who don’t care about social justice; however, the dry approach is off-putting to those who do care about social justice. In fact, I believe that all academic disciplines should address our most important issues in whatever way they can. Abstract maths is about making rigorous logical arguments, which is relevant to everything. I don’t demand that students agree with me about politics, but I do ask that they construct rigorous arguments to back up their thoughts and develop the crucial ability to analyse the logic of people they disagree with.

Maths isn’t just about numbers and equations, it is about studying different logical systems in which different arguments are valid. We can apply it to balls rolling down different hills, but we can also apply it to pressing social issues. I think we should do both, for the sake of society and to be more inclusive towards different types of student in maths education.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Eugenia Cheng*

One of the essential skills students need to master in primary school mathematics are “multiplication facts.”

What are they? What are they so important? And how can you help your child master them?

What are multiplication facts?

Multiplication facts typically describe the answers to multiplication sums up to 10×10. Sums up to 10×10 are called “facts” as it is expected they can be easily and quickly recalled. You may recall learning multiplication facts in school from a list of times tables.

The shift from “times tables” to “multiplication facts” is not just about language. It stems from teachers wanting children to see how multiplication facts can be used to solve a variety of problems beyond the finite times table format.

For example, if you learned your times tables in school (which typically went up to 12×12 and no further), you might be stumped by being asked to solve 15×8 off the top of your head. In contrast, we hope today’s students can use their multiplication facts knowledge to quickly see how 15×8 is equivalent to 10×8 plus 5×8.

The shift in terminology also means we are encouraging students to think about the connections between facts. For example, when presented only in separate tables, it is tricky to see how 4×3 and 3×4 are directly connected.

Math education has changed

In a previous piece, we talked about how mathematics education has changed over the past 30 years.

In today’s mathematics classrooms, teachers still focus on developing students’ mathematical accuracy and fast recall of essential facts, including multiplication facts.

But we also focus on developing essential problem-solving skills. This helps students form connections between concepts, and learn how to reason through a variety of real-world mathematical tasks.

Why are multiplication facts so important?

By the end of primary school, it is expected students will know multiplication facts up to 10×10 and can recall the related division fact (for example, 10×9=90, therefore 90÷10=9).

Learning multiplication facts is also essential for developing “multiplicative thinking.” This is an understanding of the relationships between quantities, and is something we need to know how to do on a daily basis.

When we are deciding whether it is better to purchase a 100g product for $3 or a 200g product for $4.50, we use multiplicative thinking to consider that 100g for $3 is equivalent to 200g for $6—not the best deal!

Multiplicative thinking is needed in nearly all math topics in high school and beyond. It is used in many topics across algebra, geometry, statistics and probability.

This kind of thinking is profoundly important. Research shows students who are more proficient in multiplicative thinking perform significantly better in mathematics overall.

In 2001, an extensive RMIT study found there can be as much as a seven-year difference in student ability within one mathematics class due to differences in students’ ability to access multiplicative thinking.

These findings have been confirmed in more recent studies, including a 2021 paper.

So, supporting your child to develop their confidence and proficiency with multiplication is key to their success in high school mathematics. How can you help?

Below are three research-based tips to help support children from Year 2 and beyond to learn their multiplication facts.

1. Discuss strategies

One way to help your child’s confidence is to discuss strategies for when they encounter new multiplication facts.

Prompt them to think of facts they already and how they can be used for the new fact.

For example, once your child has mastered the x2 multiplication facts, you can discuss how 3×6 (3 sixes) can be calculated by doubling 6 (2×6) and adding one more 6. We’ve now realized that x3 facts are just x2 facts “and one more”!

Strategies can be individual: students should be using the strategy that makes the most sense to them. So you could ask a questions such as “if you’ve forgotten 6×7, how could you work it out?” (we might personally think of 6×6=36 and add one more 6, but your child might do something different and equally valid).

This is a great activity for any quiet car trip. It can also be a great drawing activity where you both have a go at drawing your strategy and then compare. Identifying multiple strategies develops flexible thinking.

2. Help them practice

Practicing recalling facts under a friendly time crunch can be helpful in achieving what teachers call “fluency” (that is, answering quickly and easily).

A great game you could play with your children is “multiplication heads up” . Using a deck of cards, your child places a card to their forehead where you can see but they cannot. You then flip over the top card on the deck and reveal it to your child. Using the revealed card and the card on your child’s head you tell them the result of the multiplication (for example, if you flip a 2 and they have a 3 card, then you tell them “6!”).

Based on knowing the result, your child then guesses what their card was.

If it is challenging to organize time to pull out cards, you can make an easier game by simply quizzing your child. Try to mix it up and ask questions that include a range of things they know well with and ones they are learning.

Repetition and rehearsal will mean things become stored in long-term memory.

3. Find patterns

Another great activity to do at home is print some multiplication grids and explore patterns with your child.

A first start might be to give your child a blank or partially blank multiplication grid which they can practice completing.

Then, using colored pencils, they can color in patterns they notice. For example, the x6 column is always double the answer in the x3 column. Another pattern they might see is all the even answers are products of 2, 4, 6, 8, 10. They can also notice half of the grid is repeated along the diagonal.

This also helps your child become a mathematical thinker, not just a calculator.

The importance of multiplication for developing your child’s success and confidence in mathematics cannot be understated. We believe these ideas will give you the tools you need to help your child develop these essential skills.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Bronwyn Reid O’Connor and Benjamin Zunica, The Conversation

 

Not content with shapes in two or three dimensions, mathematicians like to explore objects in any number of spatial dimensions. Now they have discovered shapes of constant width in any dimension, which roll like a wheel despite not being round.

A 3D shape of constant width as seen from three different angles. The middle view resembles a 2D Reuleaux triangle

Mathematicians have reinvented the wheel with the discovery of shapes that can roll smoothly when sandwiched between two surfaces, even in four, five or any higher number of spatial dimensions. The finding answers a question that researchers have been puzzling over for decades.

Such objects are known as shapes of constant width, and the most familiar in two and three dimensions are the circle and the sphere. These aren’t the only such shapes, however. One example is the Reuleaux triangle, which is a triangle with curved edges, while people in the UK are used to handling equilateral curve heptagons, otherwise known as the shape of the 20 and 50 pence coins. In this case, being of constant width allows them to roll inside coin-operated machines and be recognised regardless of their orientation.

Crucially, all of these shapes have a smaller area or volume than a circle or sphere of the equivalent width – but, until now, it wasn’t known if the same could be true in higher dimensions. The question was first posed in 1988 by mathematician Oded Schramm, who asked whether constant-width objects smaller than a higher-dimensional sphere might exist.

While shapes with more than three dimensions are impossible to visualise, mathematicians can define them by extending 2D and 3D shapes in logical ways. For example, just as a circle or a sphere is the set of points that sits at a constant distance from a central point, the same is true in higher dimensions. “Sometimes the most fascinating phenomena are discovered when you look at higher and higher dimensions,” says Gil Kalai at the Hebrew University of Jerusalem in Israel.

Now, Andrii Arman at the University of Manitoba in Canada and his colleagues have answered Schramm’s question and found a set of constant-width shapes, in any dimension, that are indeed smaller than an equivalent dimensional sphere.

Arman and his colleagues had been working on the problem for several years in weekly meetings, trying to come up with a way to construct these shapes before they struck upon a solution. “You could say we exhausted this problem until it gave up,” he says.

The first part of the proof involves considering a sphere with n dimensions and then dividing it into 2ⁿ equal parts – so four parts for a circle, eight for a 3D sphere, 16 for a 4D sphere and so on. The researchers then mathematically stretch and squeeze these segments to alter their shape without changing their width. “The recipe is very simple, but we understood that only after all of our elaboration,” says team member Andriy Bondarenko at the Norwegian University of Science and Technology.

The team proved that it is always possible to do this distortion in such a way that you end up with a shape that has a volume at most 0.9ⁿ times that of the equivalent dimensional sphere. This means that as you move to higher and higher dimensions, the shape of constant width gets proportionally smaller and smaller compared with the sphere.

Visualising this is difficult, but one trick is to imagine the lower-dimensional silhouette of a higher-dimensional object. When viewed at certain angles, the 3D shape appears as a 2D Reuleaux triangle (see the middle image above). In the same way, the 3D shape can be seen as a “shadow” of the 4D one, and so on.  “The shapes in higher dimensions will be in a certain sense similar, but will grow in complexity as [the] dimension grows,” says Arman.

Having identified these shapes, mathematicians now hope to study them further. “Even with the new result, which takes away some of the mystery about them, they are very mysterious sets in high dimensions,” says Kalai.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Alex Wilkins*

School mathematics teaching is stuck in the past. An adult revisiting the school that they attended as a child would see only superficial changes from what they experienced themselves.

Yes, in some schools they might see a room full of electronic tablets, or the teacher using a touch-sensitive, interactive whiteboard. But if we zoom in on the details – the tasks that students are actually being given to help them make sense of the subject – things have hardly changed at all.

We’ve learnt a huge amount in recent years about cognitive science – how our brains work and how people learn most effectively. This understanding has the potential to revolutionise what teachers do in classrooms. But the design of mathematics teaching materials, such as textbooks, has benefited very little from this knowledge.

Some of this knowledge is counter-intuitive, and therefore unlikely to be applied unless done so deliberately. What learners prefer to experience, and what teachers think is likely to be most effective, often isn’t what will help the most.

For example, cognitive science tells us that practising similar kinds of tasks all together generally leads to less effective learning than mixing up tasks that require different approaches.

In mathematics, practising similar tasks together could be a page of questions each of which requires addition of fractions. Mixing things up might involve bringing together fractions, probability and equations in immediate succession.

Learners make more mistakes when doing mixed exercises, and are likely to feel frustrated by this. Grouping similar tasks together is therefore likely to be much easier for the teacher to manage. But the mixed exercises give the learner important practice at deciding what method they need to use for each question. This means that more knowledge is retained afterwards, making this what is known as a “desirable difficulty”.

Cognitive science applied

We are just now beginning to apply findings like this from cognitive science to design better teaching materials and to support teachers in using them. Focusing on school mathematics makes sense because mathematics is a compulsory subject which many people find difficult to learn.

Typically, school teaching materials are chosen by gut reactions. A head of department looks at a new textbook scheme and, based on their experience, chooses whatever seems best to them. What else can they be expected to do? But even the best materials on offer are generally not designed with cognitive science principles such as “desirable difficulties” in mind.

My colleagues and I have been researching educational designthat applies principles from cognitive science to mathematics teaching, and are developing materials for schools. These materials are not designed to look easy, but to include “desirable difficulties”.

They are not divided up into individual lessons, because this pushes the teacher towards moving on when the clock says so, regardless of student needs. Being responsive to students’ developing understanding and difficulties requires materials designed according to the size of the ideas, rather than what will fit conveniently onto a double-page spread of a textbook or into a 40-minute class period.

Switching things up

Taking an approach led by cognitive science also means changing how mathematical concepts are explained. For instance, diagrams have always been a prominent feature of mathematics teaching, but often they are used haphazardly, based on the teacher’s personal preference. In textbooks they are highly restricted, due to space constraints.

Often, similar-looking diagrams are used in different topics and for very different purposes, leading to confusion. For example, three circles connected as shown below can indicate partitioning into a sum (the “part-whole model”) or a product of prime factors.

These involve two very different operations, but are frequently represented by the same diagram. Using the same kind of diagram to represent conflicting operations (addition and multiplication) leads to learners muddling them up and becoming confused.

Number diagrams showing numbers that add together to make six and numbers that multiply to make six. Colin Foster

The “coherence principle” from cognitive science means avoiding diagrams where their drawbacks outweigh their benefits, and using diagrams and animations in a purposeful, consistent way across topics.

For example, number lines can be introduced at a young age and incorporated across many topic areas to bring coherence to students’ developing understanding of number. Number lines can be used to solve equations and also to represent probabilities, for instance.

Unlike with the circle diagrams above, the uses of number lines shown below don’t conflict but reinforce each other. In each case, positions on the number line represent numbers, from zero on the left, increasing to the right.

A number line used to solve an equation. Colin Foster

A number line used to show probability. Colin Foster

There are disturbing inequalities in the learning of mathematics, with students from poorer backgrounds underachieving relative to their wealthier peers. There is also a huge gender participation gap in maths, at A-level and beyond, which is taken by far more boys than girls.

Socio-economically advantaged families have always been able to buy their children out of difficulties by using private tutors, but less privileged families cannot. Better-quality teaching materials, based on insights from cognitive science, mitigate the impact for students who have traditionally been disadvantaged by gender, race or financial background in the learning of mathematics.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to SrideeStudio/Shutterstock

For all the students wondering why they would ever need to use the Pythagorean theorem, Katie Steckles is delighted to report on a real-world encounter.

Recently, I was building a flat-pack wardrobe when I noticed something odd in the instructions. Before you assembled the wardrobe, they said, you needed to measure the height of the ceiling in the room you were going to put it in. If it was less than 244 centimetres high, there was a different set of directions to follow.

These separate instructions asked you to build the wardrobe in a vertical orientation, holding the side panels upright while you attached them to the base. The first set of directions gave you a much easier job, building the wardrobe flat on the floor before lifting it up into place. I was intrigued by the value of 244 cm: this wasn’t the same as the height of the wardrobe, or any other dimension on the package, and I briefly wondered where that number had come from. Then I realised: Pythagoras.

The wardrobe was 236 cm high and 60 cm deep. Looking at it side-on, the length of the diagonal line from corner to corner can be calculated using Pythagoras’s theorem. The vertical and horizontal sides meet at a right angle, meaning if we square the length of each then add them together, we get the well-known “square of the hypotenuse”. Taking the square root of this number gives the length of the diagonal.

In this case, we get a diagonal length a shade under 244 cm. If you wanted to build the wardrobe flat and then stand it up, you would need that full diagonal length to fit between the floor and the ceiling to make sure it wouldn’t crash into the ceiling as it swung past – so 244 cm is the safe ceiling height. It is a victory for maths in the real world, and vindication for maths teachers everywhere being asked, “When am I going to use this?”

This isn’t the only way we can connect Pythagoras to daily tasks. If you have ever needed to construct something that is a right angle – like a corner in joinery, or when laying out cones to delineate the boundaries of a sports pitch – you can use the Pythagorean theorem in reverse. This takes advantage of the fact that a right-angled triangle with sides of length 3 and 4 has a hypotenuse of 5 – a so-called 3-4-5 triangle.

If you measure 3 units along one side from the corner, and 4 along the other, and join them with a diagonal, the diagonal’s length will be precisely 5 units, if the corner is an exact right angle. Ancient cultures used loops of string with knots spaced 3, 4 and 5 units apart – when held out in a triangle shape, with a knot at each vertex, they would have a right angle at one corner. This technique is still used as a spot check by builders today.

Engineers, artists and scientists might use geometrical thinking all the time, but my satisfaction in building a wardrobe, and finding the maths checked out perfectly, is hard to beat.

Katie Steckles is a mathematician, lecturer, YouTuber and author based in Manchester, UK. She is also puzzle adviser for New Scientist’s puzzle column, BrainTwister. Follow her @stecks

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Peter Rowlett*

It is said that the ancient Greek philosopher Pythagoras came up with the idea that musical note combinations sound best in certain mathematical ratios, but that doesn’t seem to be true.

Pythagoras has influenced Western music for millennia

An ancient Greek belief about the most pleasing combinations of musical notes – often attributed to the philosopher Pythagoras – doesn’t actually reflect the way people around the world appreciate harmony, researchers have found. Instead, Pythagoras’s mathematical arguments may merely have been taken as fact and used to assert the superiority of Western culture.

According to legend, Pythagoras found that the ringing sounds of a blacksmith’s hammers sounded most pleasant, or “consonant”, when the ratio between the size of two tools involved two integers, or whole numbers, such as 3:2.

This idea has shaped how Western musicians play chords, because the philosopher’s belief that listeners prefer music played in perfect mathematical ratios was so influential. “Consonance is a really important concept in Western music, in particular for telling us how we build harmonies,” says Peter Harrison at the University of Cambridge.

But when Harrison and his colleagues surveyed 4272 people in the UK and South Korea about their perceptions of music, their findings flew in the face of this ancient idea.

In one experiment, participants were played musical chords and asked to rate how pleasant they seemed. Listeners were found to slightly prefer sounds with an imperfect ratio. Another experiment discovered little difference in appeal between the sounds made by instruments from around the world, including the bonang, an Indonesian gong chime, which produces harmonies that cannot be replicated on a Western piano.

While instruments like the bonang have traditionally been called “inharmonic” by Western music culture, study participants appreciated the sounds the instrument and others like it made. “If you use non-Western instruments, you start preferring different harmonies,” says Harrison.

“It’s fascinating that music can be so universal yet so diverse at the same time,” says Patrick Savage at the University of Auckland, New Zealand. He says that the current study also contradicts previous research he did with some of the same authors, which found that integer ratio-based rhythms are surprisingly universal.

Michelle Phillips at the Royal Northern College of Music in Manchester, UK, points out that the dominance of Pythagorean tunings, as they are known, has been in question for some time. “Research has been hinting at this for 30 to 40 years, as music psychology has grown as a discipline,” she says. “Over the last fifteenish years, people have undertaken more work on music in the whole world, and we now know much more about non-Western pitch perception, which shows us even more clearly how complex perception of harmony is.”

Harrison says the findings tell us both that Pythagoras was wrong about music – and that music and music theory have been too focused on the belief that Western views are held worldwide. “The idea that simple integer ratios are superior could be framed as an example of mathematical justification for why we’ve got it right over here,” he says. “What our studies are showing is that, actually, this is not an inviolable law. It’s something that depends very much on the way in which you’re playing music.”

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*Credit for article given to Chris Stokel-Walker*

An unsatisfying caveat in a mathematical breakthrough discovery of a single tile shape that can cover a surface without ever creating a repeating pattern has been eradicated. The newly discovered “spectre” shape can cover a surface without repeating and without mirror images.

The pattern on the left side is made up of the “hat” shape, including reflections. The pattern on the right is made up of round-edged “spectre” shapes that repeat infinitely without reflections

David Smith et al

Mathematicians solved a decades-long mystery earlier this year when they discovered a shape that can cover a surface completely without ever creating a repeating pattern. But the breakthrough had come with a caveat: both the shape and its mirror image were required. Now the same team has discovered that a tweaked version of the original shape can complete the task without its mirror.

Simple shapes such as squares and equilateral triangles can tile a surface without gaps in a repeating pattern. Mathematicians have long been interested in a more complex version of tiling, known as aperiodic tiling, which involves using more complex shapes that never form such a repeating pattern.

The most famous aperiodic tiles were created by mathematician Roger Penrose, who in the 1970s discovered that two different shapes could be combined to create an infinite, never-repeating tiling. In March, Chaim Goodman-Strauss at the University of Arkansas and his colleagues found the “hat”, a shape that could technically do it alone, but using a left-handed and right-handed version. This was a slightly unsatisfying solution and left the question of whether a single shape could achieve the same thing with no reflections remaining.

The researchers have now tweaked the equilateral polygon from their previous research to create a new family of shapes called spectres. These shapes allow non-repeating pattern tiling using no reflections at all.

Until now, it wasn’t clear whether such a single shape, known as an einstein (from the German “ein stein” or “one stone”), could even exist. The researchers say in their paper that the previous discovery of the hat was a reminder of how little understood tiling patterns are, and that they were surprised to make another breakthrough so soon.

“Certainly there is no evidence to suggest that the hat (and the continuum of shapes to which it belongs) is somehow unique, and we might therefore hope that a zoo of interesting new monotiles will emerge in its wake,” the researchers write in their new paper. “Nonetheless, we did not expect to find one so close at hand.”

Sarah Hart at Birkbeck, University of London, says the new result is even more impressive than the original finding. “It’s very intellectually satisfying to have a solution that doesn’t need the mirror image because if you actually had real tiles then a tile and its mirror image are not the same,” she says. “With this new tile there are no such caveats.”

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*Credit for article given to Matthew Sparkes*