Category: mathematics

When choosing the perfectly sized table to do your jigsaw puzzle on, work out the area of the completed puzzle and multiply it by 1.73.

People may require a larger table if they like to lay all the pieces out at the start, rather than keeping them in the box or in piles

How large does your table need to be when doing a jigsaw puzzle? The answer is the area of the puzzle when assembled multiplied by 1.73. This creates just enough space for all the pieces to be laid flat without any overlap.

“My husband and I were doing a jigsaw puzzle one day and I just wondered if you could estimate the area that the pieces take up before you put the puzzle together,” says Madeleine Bonsma-Fisher at the University of Toronto in Canada.

To uncover this, Bonsma-Fisher and her husband Kent Bonsma-Fisher, at the National Research Council Canada, turned to mathematics.

Puzzle pieces take on a range of “funky shapes” that are often a bit rectangular or square, says Madeleine Bonsma-Fisher. To get around the variation in shapes, the pair worked on the basis that all the pieces took up the surface area of a square. They then imagined each square sitting inside a circle that touches its corners.

By considering the area around each puzzle piece as a circle, a shape that can be packed in multiple ways, they found that a hexagonal lattice, similar to honeycomb, would mean the pieces could interlock with no overlap. Within each hexagon is one full circle and parts of six circles.

They then found that the area taken up by the unassembled puzzle pieces arranged in the hexagonal pattern would always be the total area of the completed puzzle – calculated by multiplying its length by its width – multiplied by the root of 3, or 1.73.

This also applies to jigsaw puzzle pieces with rectangular shapes, seeing as these would similarly fit within a circle.

While doing a puzzle, some people keep pieces that haven’t yet been used in the box, while others arrange them in piles or lay them on a surface, the latter being Madeleine Bonsma-Fisher’s preferred method. “If you really want to lay all your pieces out flat and be comfortable, your table should be a little over twice as big as your sample puzzle,” she says.

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*Credit for article given to Chen Ly*

A pair of mathematicians studied the UK National Lottery and figured out a combination of 27 tickets that guarantees you will always win, but they tell New Scientist they don’t bother to play.

David Cushing and David Stewart calculate a winning solution

Earlier this year, two mathematicians revealed that it is possible to guarantee a win on the UK national lottery by buying just 27 tickets, despite there being 45,057,474 possible draw combinations. The pair were shocked to see their findings make headlines around the world and inspire numerous people to play these 27 tickets – with mixed results – and say they don’t bother to play themselves.

David Cushing and David Stewart at the University of Manchester, UK, used a mathematical field called finite geometry to prove that particular sets of 27 tickets would guarantee a win.

They placed each of the lottery numbers from 1 to 59 in pairs or triplets on a point within one of five geometrical shapes, then used these to generate lottery tickets based on the lines within the shapes. The five shapes offer 27 such lines, meaning that 27 tickets will cover every possible winning combination of two numbers, the minimum needed to win a prize. Each ticket costs £2.

It was an elegant and intuitive solution to a tricky problem, but also an irresistible headline that attracted newspapers, radio stations and television channels from around the world. And it also led many people to chance their luck – despite the researchers always pointing out that it was, statistically speaking, a very good way to lose money, as the winnings were in no way guaranteed to even cover the cost of the tickets.

Cushing says he has received numerous emails since the paper was released from people who cheerily announce that they have won tiny amounts, like two free lucky dips – essentially another free go on the lottery. “They were very happy to tell me how much they’d lost basically,” he says.

The pair did calculate that their method would have won them £1810 if they had played on one night during the writing of their research paper – 21 June. Both Cushing and Stewart had decided not to play the numbers themselves that night, but they have since found that a member of their research group “went rogue” and bought the right tickets – putting himself £1756 in profit.

“He said what convinced him to definitely put them on was that it was summer solstice. He said he had this feeling,” says Cushing, shaking his head as he speaks. “He’s a professional statistician. He is incredibly lucky with it; he claims he once found a lottery ticket in the street and it won £10.”

Cushing and Stewart say that while their winning colleague – who would prefer to remain nameless – has not even bought them lunch as a thank you for their efforts, he has continued to play the 27 lottery tickets. However, he now randomly permutes the tickets to alternative 27-ticket, guaranteed-win sets in case others have also been inspired by the set that was made public. Avoiding that set could avert a situation where a future jackpot win would be shared with dozens or even hundreds of mathematically-inclined players.

Stewart says there is no way to know how many people are doing the same because Camelot, which runs the lottery, doesn’t release that information. “If the jackpot comes up and it happens to match exactly one of the [set of] tickets and it gets split a thousand ways, that will be some indication,” he says.

Nonetheless, Cushing says that he no longer has any interest in playing the 27 tickets. “I came to the conclusion that whenever we were involved, they didn’t make any money, and then they made money when we decided not to put them on. That’s not very mathematical, but it seemed to be what was happening,” he says.

And Stewart is keen to stress that mathematics, no matter how neat a proof, can never make the UK lottery a wise investment. “If every single man, woman and child in the UK bought a separate ticket, we’d only have a quarter chance of someone winning the jackpot,” he says.

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*Credit for article given to Matthew Sparkes*

A mathematical game governed by simple rules throws up patterns of seemingly infinite complexity – and now a question that has puzzled hobbyists for decades has a solution.

A pattern in the Game of Life that repeats after every 19 steps

A long-standing mystery about repeating patterns in a two-dimensional mathematical game has been solved after more than 50 years with the discovery of two final pieces in the puzzle.

The result is believed to have no practical application whatsoever, but will satisfy the curiosity of the coterie of hobbyists obsessed with the Game of Life.

Invented by mathematician John Conway in 1970, the Game of Life is a cellular automaton – a simplistic world simulation that consists of a grid of “live” cells and “dead” cells. Players create a starting pattern as an input and the pattern is updated generation after generation according to simple rules.

A live cell with fewer than two neighbouring live cells is dead in the next generation; a live cell with two or three neighbouring live cells remains live; and a live cell with more than three neighbouring live cells dies. A dead cell with exactly three neighbouring live cells becomes live in the next generation. Otherwise, it remains dead.

These rules create evolving patterns of seemingly infinite complexity that throw up three types of shape: static objects that don’t change; “oscillators”, which form a repeating but stationary pattern; and “spaceships”, which repeat but also move across the grid.

One of the enduring problems in Game of Life research is whether there are oscillators with every “period”: ones that repeat every two steps, every three steps and so on, to infinity. There was a strong clue that this would be true when mathematician David Buckingham designed a technique that could create oscillators with any period above 57, but there were still missing oscillators for some smaller numbers.

Now, a team of hobbyists has filled those last remaining gaps by publishing a paper that describes oscillators with periods of 19 and 41 – the final missing shapes.

One member of the team, Mitchell Riley at New York University Abu Dhabi, works on the problem as a hobby alongside his research in a quantum computing group. He says there are lots of methods to generate new oscillators, but no way has been found to create them with specific periods, meaning that research in this area is a game of chance. “It’s just like playing darts – we’ve just never hit 19, and we’ve never hit 41,” he says.

Riley had been scouring lists of known shapes that consist of two parts, a hassler and a catalyst. Game of Life enthusiasts coined these terms for static shapes – catalysts – that contain a changing shape inside – a hassler. The interior reacts to the exterior, but leaves it unchanged, and together they form an oscillator of a certain period. Riley’s contribution was writing a computer program to discover potentially useful catalysts.

“The stars have to align,” he says. “You need the reaction in the middle to not destroy the thing on the outside, and the reaction in the middle, just by chance, to return to its original state in one of these new periods.”

Riley says that there are no applications known for this research and that he was drawn to the problem by “pure curiosity”.

Susan Stepney at the University of York, UK, says the work demonstrates some “extremely clever and creative techniques”, but it certainly isn’t the final conclusion of research on Conway’s creation.

“I don’t think work on Game of Life will ever be complete,” says Stepney. “The system is computationally universal, so there is always more behaviour to find, and it is seemingly so simple to describe, but so complex in its behaviour, that it remains fascinating to many.”

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*Credit for article given to Matthew Sparkes*

Atomic shapes are so simple that they can’t be broken down any further. Mathematicians are trying to build a “periodic table” of these shapes, and they hope artificial intelligence can help.

Mathematicians attempting to build a “periodic table” of shapes have turned to artificial intelligence for help – but say they don’t understand how it works or whether it can be 100 per cent reliable.

Tom Coates at Imperial College London and his colleagues are working to classify shapes known as Fano varieties, which are so simple that they can’t be broken down into smaller components. Just as chemists arranged elements in the periodic table by their atomic weight and group to reveal new insights, the researchers hope that organising these “atomic” shapes by their various properties will help in understanding them.

The team has assigned each atomic shape a sequence of numbers derived from features such as the number of holes it has or the extent to which it twists around itself. This acts as a bar code to identify it.

Coates and his colleagues have now created an AI that can predict certain properties of these shapes from their bar code numbers alone, with an accuracy of 98 per cent – suggesting a relationship that some mathematicians intuitively thought might be real, but have found impossible to prove.

Unfortunately, there is a vast gulf between demonstrating that something is very often true and mathematically proving that it is always so. While the team suspects a one-to-one connection between each shape and its bar code, the mathematics community is “nowhere close” to proving this, says Coates.

“In pure mathematics, we don’t regard anything as true unless we have an actual proof written down on a piece of paper, and no advances in our understanding of machine learning will get around this problem,” says team member Alexander Kasprzyk at the University of Nottingham, UK.

Even without a proven link between the Fano varieties and bar codes, Kasprzyk says that the AI has let the team organise atomic shapes in a way that begins to mimic the periodic table, so that when you read from left to right, or up and down, there seem to be generalisable patterns in the geometry of the shapes.

“We had no idea that would be true, we had no idea how to begin doing it,” says Kasprzyk. “We probably would still not have had any idea about this in 50 years’ time. Frankly, people have been trying to study these things for 40 years and failing to get to a picture like this.”

The team hopes to refine the model to the point where missing spaces in its periodic table could point to the existence of unknown shapes, or where clustering of shapes could lead to logical categorisation, resulting in a better understanding and new ideas that could create a method of proof. “It clearly knows more things than we know, but it’s so mysterious right now,” says team member Sara Veneziale at Imperial College London.

Graham Niblo at the University of Southampton, UK, who wasn’t involved in the research, says that the work is akin to forming an accurate picture of a cello or a French horn just from the sound of a G note being played – but he stresses that humans will still need to tease understanding from the results provided by AI and create robust and conclusive proofs of these ideas.

“AI has definitely got uncanny abilities. But in the same way that telescopes didn’t put astronomers out of work, AI doesn’t put mathematicians out of work,” he says. “It just gives us a new tool that allows us to explore parts of the mathematical landscape that were out of reach, or, like a microscope, that were too obscure for us to notice with our current understanding.”

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*Credit for article given to Matthew Sparkes *

Until recently, working out how three objects can stably orbit each other was nearly impossible, but now mathematicians have found a record number of solutions.

The motion of three objects is more complex than you might think

The question of how three objects can form a stable orbit around each other has troubled mathematicians for more than 300 years, but now researchers have found a record 12,000 orbital arrangements permitted by Isaac Newton’s laws of motion.

While mathematically describing the movement of two orbiting bodies and how each one’s gravity affects the other is relatively simple, the problem becomes vastly more complex once a third object is added. In 2017, researchers found 1223 new solutions to the three-body problem, doubling the number of possibilities then known. Now, Ivan Hristov at Sofia University in Bulgaria and his colleagues have unearthed more than 12,000 further orbits that work.

The team used a supercomputer to run an optimised version of the algorithm used in the 2017 work, discovering 12,392 new solutions. Hristov says that if he repeated the search with even more powerful hardware he could find “five times more”.

All the solutions found by the researchers start with all three bodies being stationary, before entering freefall as they are pulled towards each other by gravity. Their momentum then carries them past each other before they slow down, stop and are attracted together once more. The team found that, assuming there is no friction, this pattern would repeat infinitely.

Solutions to the three-body problem are of interest to astronomers, as they can describe how any three celestial objects – be they stars, planets or moons – can maintain a stable orbit. But it remains to be seen how stable the new solutions are when the tiny influences of additional, distant bodies and other real-world noise are taken into account.

“Their physical and astronomical relevance will be better known after the study of stability – it’s very important,” says Hristov. “But, nevertheless – stable or unstable – they are of great theoretical interest. They have a very beautiful spatial and temporal structure.”

Juhan Frank at Louisiana State University says that finding so many solutions in a precise set of conditions will be of interest to mathematicians, but of limited application in the real world.

“Most, if not all, require such precise initial conditions that they are probably never realised in nature,” says Frank. “After a complex and yet predictable orbital interaction, such three-body systems tend to break into a binary and an escaping third body, usually the least massive of the three.”

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*Credit for article given to Matthew Sparkes *

Exciting a brain region using electrical noise stimulation can help improve mathematical learning in those who struggle with the subject, according to a new study from the Universities of Surrey and Oxford, Loughborough University, and Radboud University in The Netherlands.

During this unique study, published in PLOS Biology, researchers investigated the impact of neurostimulation on learning. Despite the growing interest in this non-invasive technique, little is known about the neurophysiological changes induced and the effect it has on learning.

Researchers found that electrical noise stimulation over the frontal part of the brain improved the mathematical ability of people whose brain was less excited (by mathematics) before the application of stimulation. No improvement in mathematical scores was identified in those who had a high level of brain excitation during the initial assessment or in the placebo groups. Researchers believe that electrical noise stimulation acts on the sodium channels in the brain, interfering with the cell membrane of the neurons, which increases cortical excitability.

Professor Roi Cohen Kadosh, Professor of Cognitive Neuroscience and Head of the School of Psychology at the University of Surrey who led this project, said, “Learning is key to everything we do in life—from developing new skills, such as driving a car, to learning how to code. Our brains are constantly absorbing and acquiring new knowledge.

“Previously, we have shown that a person’s ability to learn is associated with neuronal excitation in their brains. What we wanted to discover in this case is if our novel stimulation protocol could boost, in other words excite, this activity and improve mathematical skills.”

For the study, 102 participants were recruited, and their mathematical skills were assessed through a series of multiplication problems. Participants were then split into four groups including a learning group exposed to high-frequency random electrical noise stimulation and an overlearning group in which participants practiced the multiplication beyond the point of mastery with high-frequency random electrical noise stimulation.

The remaining two groups consisted of a learning and overlearning group but they were exposed to a sham (i.e., placebo) condition, an experience akin to real stimulation without applying significant electrical currents. EEG recordings were taken at the beginning and at the end of the stimulation to measure brain activity.

Dr. Nienke van Bueren, from Radboud University, who led this work under Professor Cohen Kadosh’s supervision, said, “These findings highlight that individuals with lower brain excitability may be more receptive to noise stimulation, leading to enhanced learning outcomes, while those with high brain excitability might not experience the same benefits in their mathematical abilities.”

Professor Cohen Kadosh adds, “What we have found is how this promising neurostimulation works and under which conditions the stimulation protocol is most effective. This discovery could not only pave the way for a more tailored approach in a person’s learning journey but also shed light on the optimal timing and duration of its application.”

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Credit of the article given to University of Surrey

Statistical methods can evaluate whether pivotal military events, like the Battle of Jutland, American involvement in the Vietnam war or the nuclear arms race, could’ve turned out otherwise, according to a new book.

Military historical narratives and statistical modeling bring fresh perspectives to the fore in a ground-breaking text, “Quantifying Counterfactual Military History,” by Brennen Fagan, Ian Horwood, Niall MacKay, Christopher Price and Jamie Wood, a team of historians and mathematicians.

The authors explain, “In writing history, it must always be remembered that a historical fact is simply one of numberless possibilities until the historical actor moves or an event occurs, at which point it becomes real. To understand the one-time possibility that became evidence we must also understand the possibilities that remained unrealized.”

Re-examining the battlefield

Midway through the First World War, Britain and Germany were locked in a technologically driven arms race which culminated in the Battle of Jutland in 1916. By that time, both nations had built over 50 dreadnoughts—speedy, heavily-armored, turbine-powered “all-big-gun” warships.

In the context of a battle that Britain couldn’t afford to lose, Winston Churchill was quoted as saying that the British commander John Jellicoe was “the only man on either side who could lose the war in an afternoon.”

Both nations, at various points in time, have claimed victory in the Battle of Jutland, and there is no consensus on who “won.” Using mathematical modeling, “Quantifying Counterfactual Military History” probes whether the Germans could have achieved a decisive victory.

The five scholars note, “This reconstructive battling enables us to put some level of statistical insight into multiple realizations of a key phase of Jutland. The model is crude and laden with assumptions—as are all wargames—but, unlike in a wargame, our goal is simply to understand what is plausible and what is not.”

Understanding nuclear deterrence

Counterfactual reasoning is positioned centrally when it comes to the fraught history of nuclear deterrence. By the 1980s, the nuclear arms race between the US and the Soviet Union had already spanned three decades and 1983 would bring a crisis less well-known than the Cuban missile crisis of 1962.

The authors draw attention to the peak of intensity in November 1983 during the so-called “Second Cold War.” A NATO “command post” exercise in Western Europe—known as Able Archer—was created to test communications in the event of nuclear war.

The Soviets, however—likely the result of faulty intelligence gathering—believed that an attack was imminent with the NATO exercise interpreted as the first phase. “Quantifying Counterfactual Military History” highlights the example as one where each side placed themselves in dangerous counterfactual mindsets.

“Mutual misapprehension in 1983 continued a long tradition of misunderstanding which had always created catastrophic potential for war, now based consciously and unconsciously on game theory and its erroneous assumption that rational actors were guided by accurate information,” the authors explain. “In this way they stumbled towards a war that neither had willed.”

Embrace the alternative

“Quantifying Counterfactual Military History” uses case studies of Jutland, Able Archer, the Battle of Britain and the Vietnam War to appraise long-established narratives around military events and examine the probabilities of the events that took place alongside the potential for alternative outcomes.

The book’s authors, however, take a restrained approach to counterfactual theory, one that acknowledges and considers why some events—including the actions of individuals or the rise of institutions—are more important than others and can be considered “critical junctures.” They understand this as very different from the arbitrary, loosely substantiated suppositions made by “exuberant” counterfactuals.

They say, “We can never be certain of the existence of critical junctures, or of the grounds of their criticality, but ‘restrained’ counterfactuals, if done with multiple perspectives and sufficient thoroughness, can surely make a distinctive contribution to the literature.”

The book is underpinned by an inter-disciplinary method which combines historical narrative and statistical data and analysis, offering both quantitative and qualitative rigor.

They explain, “This study has taken us in directions which are not common in academic collaboration, but which we hope demonstrate that collaborative research exploring what had been dead ground between the sciences and the humanities is long overdue.”

Rather than attempt to merely reinvent the past, “Quantifying Counterfactual Military History” calls attention to the dynamism inherent in historical practice and offers another tool for understanding historical actors, the decisions they made and the futures they shaped.

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Credit of the article given to Taylor & Francis

There is more to where maths came from than the ancient Greeks. From calculus to the theorem we credit to Pythagoras, so much of our knowledge comes from other places, including ancient China, India and the Arabian peninsula, says Kate Kitagawa.

The history of mathematics has an image problem. It is often presented as a meeting of minds among ancient Greeks who became masters of logic. Pythagoras, Euclid and their pals honed the tools for proving theorems and that led them to the biggest results of ancient times. Eventually, other European greats like Leonhard Euler and Isaac Newton came along and made maths modern, which is how we got to where we are today.

But, of course, this telling is greatly distorted. The history of maths is far richer, more chaotic and more diverse than it is given credit for. So much of what is now incorporated into our global knowledge comes from other places, including ancient China, India and the Arabian peninsula.

Take “Pythagoras’s” theorem. This is the one that says that in right-angled triangles, the square of the longest side is the sum of the square of the other two sides. The ancient Greeks certainly knew about this theorem, but so too did mathematicians in ancient Babylonia, Egypt, India and China.

In fact, in the 3rd century AD, Chinese mathematician Liu Hui added a proof of the theorem to the already old and influential book The Nine Chapters on the Mathematical Art. His version includes the earliest written statement of the theorem that we know of. So perhaps we should really call it Liu’s theorem or the gougu theorem as it was known in China.

The history of maths is filled with tales like this. Ideas have sprung up in multiple places at multiple times, leaving room for interpretation as to who should get the credit. As if credit is something that can’t be split.

As a researcher on the history of maths, I had come across examples of distorted views, but it was only when working on a new book, The Secret Lives of Numbers, that I found out just how pervasive they are. Along with my co-author, New Scientist‘s Timothy Revell, we found that the further we dug, the more of the true history of maths there was to uncover.

Another example is the origins of calculus. This is often presented as a battle between Newton and Gottfried Wilhelm Leibniz, two great 17th-century European mathematicians. They both independently developed extensive theories of calculus, but missing from the story is how an incredible school in Kerala, India, led by the mathematician Mādhava, hit upon some of the same ideas 300 years before.

The idea that the European way of doing things is superior didn’t originate in maths – it came from centuries of Western imperialism – but it has infiltrated it. Maths outside ancient Greece has often been put to one side as “ethnomathematics”, as if it were a side story to the real history.

In some cases, history has also distorted legacies. Sophie Kowalevski, who was born in Moscow in 1850, is now a relatively well-known figure. She was a fantastic mathematician, known for tackling a problem she dubbed a “mathematical mermaid” for its allure. The challenge was to describe mathematically how a spinning top moves, and she made breakthroughs where others had faltered.

During her life, she was constantly discouraged from pursuing maths and often had to work for free, collecting tuition money from her students in order to survive. After her death, biographers then tainted her life, painting her as a femme fatale who relied on her looks, implying she effectively passed off others’ work as her own. There is next to no evidence this is true.

Thankfully, historians of mathematics are re-examining and correcting the biases and stereotypes that have plagued the field. This is an ongoing process, but by embracing its diverse and chaotic roots, the next chapters for maths could be the best yet.

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*Credit for article given to Kate Kitagawa *

Buying a specific set of 27 tickets for the UK National Lottery will mathematically guarantee that you win something.

Buying 27 tickets ensures a win in the UK National Lottery

You can guarantee a win in every draw of the UK National Lottery by buying just 27 tickets, say a pair of mathematicians – but you won’t necessarily make a profit.

While there are many variations of lottery in the UK, players in the standard “Lotto” choose six numbers from 1 to 59, paying £2 per ticket. Six numbers are randomly drawn and prizes are awarded for tickets matching two or more.

David Cushing and David Stewart at the University of Manchester, UK, claim that despite there being 45,057,474 combinations of draws, it is possible to guarantee a win with just 27 specific tickets. They say this is the optimal number, as the same can’t be guaranteed with 26.

The proof of their idea relies on a mathematical field called finite geometry and involves placing each of the numbers from 1 to 59 in pairs or triplets on a point within one of five geometrical shapes, then using these to generate lottery tickets based on the lines within the shapes. The five shapes offer 27 such lines, meaning that 27 tickets bought using those numbers, at a cost of £54, will hit every possible winning combination of two numbers.

The 27 tickets that guarantee a win on the UK National Lottery

Their research yielded a specific list of 27 tickets (see above), but they say subsequent work has shown that there are two other combinations of 27 tickets that will also guarantee a win.

“We’ve been thinking about this problem for a few months. I can’t really explain the thought process behind it,” says Cushing. “I was on a train to Manchester and saw this [shape] and that’s the best logical [explanation] I can give.”

Looking at the winning numbers from the 21 June Lotto draw, the pair found their method would have won £1810. But the same numbers played on 1 July would have matched just two balls on three of the tickets – still a technical win, but giving a prize of just three “lucky dip” tries on a subsequent lottery, each of which came to nothing.

Stewart says proving that 27 tickets could guarantee a win was the easiest part of the research, while proving it is impossible to guarantee a win with 26 was far trickier. He estimates that the number of calculations needed to verify that would be 10¹⁶⁵, far more than the number of atoms in the universe. “There’d be absolutely no way to brute force this,” he says.

The solution was a computer programming language called Prolog, developed in France in 1971, which Stewart says is the “hero of the story”. Unlike traditional computer languages where a coder sets out precisely what a machine should do, step by step, Prolog instead takes a list of known facts surrounding a problem and works on its own to deduce whether or not a solution is possible. It takes these facts and builds on them or combines them in order to slowly understand the problem and whittle down the array of possible solutions.

“You end up with very, very elegant-looking programs,” says Stewart. “But they are quite temperamental.”

Cushing says the research shouldn’t be taken as a reason to gamble more, particularly as it doesn’t guarantee a profit, but hopes instead that it encourages other researchers to delve into using Prolog on thorny mathematical problems.

A spokesperson from Camelot, the company that operates the lottery, told New Scientist that the paper made for “interesting reading”.

“Our approach has always been to have lots of people playing a little, with players individually spending small amounts on our games,” they say. “It’s also important to bear in mind that, ultimately, Lotto is a lottery. Like all other National Lottery draw-based games, all of the winning Lotto numbers are chosen at random – any one number has the same and equal chance of being drawn as any other, and every line of numbers entered into a draw has the same and equal chance of winning as any other.”

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*Credit for article given to Matthew Sparkes*

Putting off college math could improve the likelihood that students remain in college. But that may only be true as long as students don’t procrastinate more than one year. This is what colleagues and I found in a study published in 2023 of 1,119 students at a public university for whom no remedial coursework was required during their first year.

Enrolling in a math course during the first semester of college resulted in students being four times more likely to drop out. Although delayed enrollment in a math course had benefits in the first year, its advantages vanished by the end of the second year. In our study, almost 40% of students who postponed the course beyond a year did not attempt it at all and failed to obtain a degree within six years.

Why it matters

Nearly 1.7 million students who recently graduated from high school will immediately enroll in college. Math is a requirement for most degrees, but students aren’t always ready to do college-level math. By putting off college math for a year, it gives students time to adjust to college and prepare for more challenging coursework.

Approximately 40% of four-year college students must first take a remedial math course. This can extend the time it takes to graduate and increase the likelihood of dropping out. Our study did not apply to students who need remedial math.

For students who do not require remedial courses, some delay can be beneficial, but students’ past experiences in math can lead to avoidance of math courses. Many students experience math anxiety. Procrastination can be an avoidance strategy for managing fears about math. The fear of math for students may be a more significant barrier than their performance.

It is estimated that at least 17% of the population will likely experience high levels of math anxiety. Math anxiety can lead to a drop in math performance. It can also lead to avoiding majors and career paths involving math.

Our study fills the void in research on the effects of how soon students take college-level math courses. It also supports prior evidence that students benefit from a mix of coursework that is challenging yet not overwhelming as they transition to college.

What still isn’t known

We believe colleges need to better promote student confidence in math by examining how student success courses can reduce math anxiety. Student success courses provide students with study skills, note-taking skills, goal setting, time management and stress management, as well as career and financial decision making to support the transition to college. Although student success courses are a proven practice that help students stick with college, rarely do these courses address students’ fear of math.

Students are at the greatest risk of dropping out of college during their first year. Advisors play a crucial role in providing students with resources for success. This includes recommendations on what courses to take and when to take them. More research is also needed about how advisors can effectively communicate the impact of when math is taken by students.

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Credit of the article given to Forrest Lane, The Conversation