Author: IMC

A large team of researchers affiliated with multiple institutions across Europe, has found evidence backing up work by Persi Diaconis in 2007 in which he suggested tossed coins are more likely to land on the same side they started on, rather than on the reverse. The team conducted experiments designed to test the randomness of coin flipping and posted their results on the arXiv preprint server.

For many years, the coin toss (or flip) has represented a fair way to choose between two options—which side of a team goes first, for example, who wins a tied election, or gets to eat the last brownie. Over the years, many people have tested the randomness of coin tossing and most have found it to be as fair as expected—provided a fair coin is used.

But, Diaconis noted, such tests have only tested the likelihood that a fair coin, once flipped, has an equal chance of landing on heads or tails. They have not tested the likelihood of a fair coin landing with the same side up as that when it was flipped. He suggested that due to precession, a coin flipped into the air spends more time there with its initial side facing up, making it more likely to end up that way, as well. He suggested that the difference would be slight, however—just 1%. In this new effort, the research team tested Diaconis’ ideas.

The experiment involved 48 people flipping coins minted in 46 countries (to prevent design bias) for a total of 350,757-coin flips. Each time, the participants noted whether the coin landed with the same side up as when it was launched. The researchers found that Diaconis was right—there was a slight bias. They found the coin landed with the same side up as when it was launched 50.8% of the time. They also found there was some slight variation in percentages between different individuals tossing coins.

The team concludes that while the bias they found is slight, it could be meaningful if multiple coin tosses are used to determine an outcome—for example, flipping a quarter 1,000 times and betting $1 each time (with winnings of 0 or 2$ each round) should result in an average overall win of $19.

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Credit of the article given to Bob Yirka , Phys.org

Researchers Jayne Spiller and Camilla Gilmore at the Center for Mathematical Cognition, University of Loughborough, U.K., have investigated the intersection of sleep and mathematical memory, finding that sleep after learning improves recall.

In their paper, “Positive impact of sleep on recall of multiplication facts,” published in Royal Society Open Science, the duo investigated whether learning complex multiplication problems before sleep would benefit recall compared to learning them during wakefulness to understand how sleep affects the memory of mathematical facts, specifically multiplication tables.

The study involved 77 adult participants aged 18 to 40 from the U.K. Each participant learned complex multiplication problems in two conditions: before sleep (sleep learning) and in the morning (wake learning). Participants completed online sessions where they learned new complex multiplication problems or were tested on previously learned material. Learning sessions included both untimed and timed trials.

Participants had better recall in the sleep learning condition than in the wake learning condition, with a moderate effect size. Even when participants had varying learning abilities, the sleep learning condition showed a beneficial effect on recall, with a smaller effect size.

Mathematical proficiency of the participants, as measured by accuracy in simple multiplication problems, was associated with learning scores but not with the extent of sleep-related benefit for recall.

The study highlights the potential educational implications of leveraging sleep-related benefits for learning. The positive impact of sleep on the recall of complex multiplication problems could be particularly useful for children learning multiplication tables or other math memorization skills, though it would be interesting to see how well a bedtime math lesson would be received.

While the authors suggest that sleep conferred the additional benefit on recall compared with learning during the daytime, the mechanisms by which encoding takes place are possibly enforced by a lack of continued external inputs. The authors point out this limitation of a lack of other comparative stimuli with a similar complexity of encoding to conclusively demonstrate in their study the specificity of sleep-related benefits on recall.

Asleep, the brain may be locking in the new learning because it has no other competition.

In contrast, an awake brain may be confronted with conversations, media reading or viewing and even other classes packed with learning material. This competition for memory encoding in the waking brain could be the cause of the memory differences seen in the study, though outside of recommending multi-hour meditation sessions between classes the likelihood of finding an alternative to sleep on memory may be limited.

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Credit of the article given to P by Justin Jackson, Phys.org

Like fingerprints, a firearm’s discarded shell casings have unique markings. This allows forensic experts to compare casings from a crime scene with those from a suspect’s gun. Finding and reporting a mismatch can help free the innocent, just as a match can incriminate the guilty.

But a new study from Iowa State University researchers reveals mismatches are more likely than matches to be reported as “inconclusive” in cartridge-case comparisons.

“Firearms experts are failing to report evidence that’s favourable to the defense, and it has to be addressed and corrected. This is a terrible injustice to innocent people who are counting on expert examiners to issue a report showing that their gun was not involved but instead are left defenseless by a report that says the result was inconclusive,” says Gary Wells, an internationally recognized pioneer and scholar in eyewitness memory research.

The Distinguished Professor Emeritus co-authored the paper with Andrew Smith, associate professor of quantitative psychology. Smith studies memory, judgment and decision-making and is affiliated with both the Cognitive Psychology Program and the Psychology and Law Research group at Iowa State.

The two researchers pulled a dataset from a previously published experiment involving 228 firearms examiners and 1,811 cartridge-case comparisons. Overall, the participants were highly accurate in determining whether casings from a common firearm matched or mismatched. But when Smith and Wells applied a well-established mathematical model to the data, they found 32% of actual mismatch trials were reported as inconclusive compared to 1% of actual match trials.

“If the 16% of inconclusive reports lined up more evenly across actual matches and non-matches, we could chalk it up to human error. But the asymmetry, combined with the near-perfect performance of examiners, indicated something else was going on. They almost certainly knew that most of the cases they called inconclusive were actual mismatches,” says Smith.

Asking the wrong question

The researchers say a flawed response scale could help explain the dissociation between what examiners know and what they report.

Currently, the Association of Firearm and Tool Mark Examiners’ Conclusion Scale asks forensic firearms experts whether the crime-scene casings and casings from the suspect’s gun are from the same source. Smith and Wells say the problem with the “source” question is that it’s possible for a mismatch to be attributable to an altered firearm or degraded evidence.

With these possible explanations, Smith and Wells say some examiners might take the position that it is never appropriate to call something a mismatch and instead default to calling the results inconclusive.

“Instead of asking examiners to make source determinations, examiners should simply be asked if the shell casings from the suspect’s gun match the casings found at the crime scene. Asking if the casings match or not and to what degree could provide more transparency,” says Smith.

Questions about alterations and degradation could be asked separately, Smith adds.

Wells emphasizes that until the response scale is fixed, defense lawyers should cross-examine forensic firearms experts who claim inconclusive results. They need to “show their work,” he says. Wells also recommends getting a second opinion if the cartridge-case comparison report comes back as inconclusive.

Bias in the lab

The researchers say another possible explanation for calling a result inconclusive when it’s actually a mismatch is “adversarial allegiance bias.”

“Most forensic firearm examiners and their labs are retained by the prosecution or police departments,” says Smith. “Some examiners might render reports that are inconclusive despite the mismatch because they don’t want to hurt the side that’s essentially their employer.”

Smith and Wells say this type of bias can also occur at the lab level. They point to survey data showing some labs have policies that do not allow examiners to report mismatches.

“It’s hard to get rid of bias but fixing the response scale would go a long way in solving the problem,” says Wells. “In the meantime, there are likely past cases that need relitigated.”

The researchers underscore that forensic science needs to be proficient in not just incriminating the guilty but also in freeing the innocent from suspicion. Minimizing bias and improving transparency in cartridge-case comparisons will help create a more fair and efficient criminal justice system.

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Credit of the article given to Iowa State University

In a remarkable breakthrough in the field of mathematical science, Professor Kyudong Choi from the Department of Mathematical Sciences at UNIST has provided an irrefutable proof that certain spherical vortices exist in a stable state. This discovery holds significant implications for predicting weather anomalies and advancing weather prediction technologies. The research is published in the journal Communications on Pure and Applied Mathematics.

A vortex is a rotating region of fluid, such as air or water, characterized by intense rotation. Common examples include typhoons and tornadoes frequently observed in news reports. Professor Choi’s mathematical proof establishes the stability of specific types of vortex structures that can be encountered in real-world fluid flows.

The study builds upon the foundational Euler equation formulated by Leonhard Euler in 1757 to describe the flow of eddy currents. In 1894, British mathematician M. Hill mathematically demonstrated that a ball-shaped vortex could maintain its shape indefinitely while moving along its axis.

Professor Choi’s research confirms that Hill’s spherical vortex maximizes kinetic energyunder certain conditions through the application of variational methods. By incorporating functional analysis and partial differential equation theory from mathematical analysis, this study extends previous investigations on two-dimensional fluid flows to encompass three-dimensional fluid dynamics with axial symmetry conditions.

One notable feature identified by Hill is the presence of strong upward airflow at the front of the spherical vortex—an attribute often observed in phenomena like typhoons and tornadoes. Professor Choi’s findings serve as a starting point for further studies involving measurements related to residual time associated with these ascending air currents.

“Research on vortex stability has gained international attention,” stated Professor Choi. “And it holds long-term potential for advancements in today’s weather forecasting technology.”

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Credit of the article given to JooHyeon Heo, Ulsan National Institute of Science and Technology

Tens of thousands of handwritten pages by one of the 20th century’s greatest mathematicians, Alexander Grothendieck, many of which the eccentric genius penned while living as a hermit, were unveiled in France on Friday.

The unpublished manuscripts, which veer from math to metaphysics, autobiography and even long musings on Satan, offer a unique insight into the enigmatic mind of the French mathematician, according to experts at the Paris library where they were donated.

Grothendieck, who died aged 86 in 2014, is considered by some to have revolutionized the field of mathematics in the way that Einstein did for physics. His work on algebraic geometry earned him the 1966 Fields Medal, known as the Nobel prize of the math world.

At that time Grothendieck was already a radical environmentalist and pacifist. But he withdrew from the world almost entirely in the early 1990s, in part to focus on what he referred to as his “scribblings”.

While living as a hermit in the southern French village of Lasserre he frantically wrote “Reflections on Life and the Cosmos,” one of the two main works added to the collection of the National Library of France (BnF) on Friday.

The massive tome includes 30,000 pages across 41 different volumes covering science, philosophy and psychology—all densely scribbled with a fountain pen.

The second work, “The Key to Dreams or Dialogue with the Good Lord,” is a typed manuscript in which he explores the interpretation of dreams.

These pages, which have previously circulated online, were written between 1987-1988.

‘Completely cut ties’

At that time, Grothendieck remained a professor at the University of Montpellier but had largely withdrawn from the mathematical community.

He became a recluse when he moved to Lasserre.

“He completely cut ties with his family, we could no longer communicate with him,” his daughter Johanna Grothendieck told AFP.

“When we sent him a letter, it was returned to sender,” said Johanna, a 64-year-old ceramic artist who traveled from southwest France to attend the ceremony at the library.

“Writing was his main activity,” she added.

Towards the end of Grothendieck’s life, a neighbour told his family that his health was deteriorating.

Johanna and one of her brothers were finally able to visit their father. It was than that they discovered “Reflections on Life and the Cosmos,” which was meticulously catalogued in his library.

In his 1997 will, Grothendieck left the early sections of the tome to the BnF. Now his children have donated the rest.

“It was an extremely important work in his eyes. He even wanted to create a foundation to look after it,” Johanna Grothendieck said.

‘Ghosts of his past’

Jocelyn Monchamp, a curator an the BnF, said the manuscripts were unique because they covered so many subjects at the same time yet formed a whole with “undeniable literary qualities”.

This is particularly the case for the autobiographical volume “Harvest and Sowing”, which depicts the author “in a metaphysical retreat,” she said.

Monchamp has spent a month poring over the writing, trying to decipher the dense fountain pen text.

“I became used to it,” she said, adding that at least Grothendieck methodically wrote the numbers and dates on all the pages.

In one of the sections, “Structures of the Psyche,” enigmatic diagrams translate psychology into the language of algebra.

In another, “The Problem of Evil,” Grothendieck muses over 15,000 pages on metaphysics and Satan.

One gets the feeling of a man “overtaken by the ghosts of his past,” Johanna Grothendieck said.

The mathematician’s father fled Germany during World War II, only to be handed by the Vichy France government to the Nazis and die at the Auschwitz concentration camp. Experts expect it will take some time to fully understand Grothendieck’s writing. On Friday, the collection joined the manuscript department of the BnF, where it will only be accessible to researchers.

During the donation ceremony, one of the volumes was placed in a glass case next to a manuscript by ancient Greek mathematician Euclid, considered the father of geometry.

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Credit of the article given to Juliette Collen

Understanding how children learn to count can have profound impacts on the kinds of instructional materials used in the classroom. And the way those materials are designed can shape the strategies children use to learn, according to a new paper led by Concordia researchers.

Writing in the journal School Science and Mathematics, the authors study how young children, mostly in the first grade, used a hundreds table to perform age-appropriate counting tasks. Hundreds tables, as the name suggests, are charts divided into rows and columns of 10, with each square containing a number from one to 100. The researchers discovered that the children who counted left-to-right, top-to-bottom outperformed children who counted left-to-right, bottom-to-top.

In this study, children used the tables on a screen to solve addition problems. One group of children used a top-down table, where the top left corner was marked 1 and bottom left corner was marked 100. Another group of children used a bottom-up, where one occupied the bottom left and 100 the top right. A third group of children used a bottom-up table with a visual cue of a cylinder next to it. The cylinder was designed to show the “up-is-more” relation as it filled with water when the numbers increased when moving up in the table.

“We found that children using the top-down chart used a more sophisticated strategy of counting by 10 and moving vertically, rather than using the more simplistic strategy of counting by one and moving horizontally,” says Vera Wagner. She co-authored the paper with Helena Osana, a professor in the Department of Education in the Faculty of Arts and Science and Jairo Navarrete-Ulloa of O’Higgins University in Chile.

The authors believe the benefits of the top-down table could be related to the way children learn to read and that they are applying the same approach to base-ten concepts.

“We were working with young children, so reading instruction is likely at the forefront of their attention,” says Wagner, who now teaches elementary students at a Montreal-area school. “The structure of moving in that particular way might be more ingrained.”

The power of spatial configuration

Osana notes that the practice of counting by 10s rather than by ones—which is a more efficient method of arriving at the same answer—is an example of unitizing, in which multiples of one unit form a new unit representing a larger number.

“From a theoretical perspective, the study shows that the spatial configuration of instructional materials can actually support this more sophisticated understanding of numbers and the unitizing aspect that goes along with it,” she says.

While the researchers are not suggesting children will automatically gravitate toward the top-down chart under every circumstance, they do think the study’s results provide educators with a sense of the ways their students process numbers and addition.

“It is important for teachers to be aware of how children are thinking about the tools we are giving them,” says Osana, principal investigator of the Mathematics Teaching and Learning Lab. “We are not saying that teachers have to use the top-down hundreds chart every time, but they should think about the strategies their students are using and why they use them with one particular instructional tool and not another.”

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Credit of the article given to Patrick Lejtenyi, Concordia University

It seems there’s a five-letter word describing what many players of the wildly popular Wordle puzzle do daily as they struggle to find a target word within six tries.

According to one mathematics expert, that word is “cheat.”

James P. Dilger, who by day is professor emeritus at Stony Brook University in New York specializing in the mechanisms of anesthetic action, and by night is a Wordle junkie, says the numbers behind published Wordle success rates don’t quite add up.

Wordle was developed by a software engineerto pass the time during the early days of COVID restrictions. Players must determine a target five-letter word in six or fewer attempts. With each guess, the player is provided with three bits of information: correct letters in the correct position are displayed in green, correct letters placed in incorrect spots are displayed in yellow, and incorrect letters are displayed in black.

In the beginning, Wordle was played mainly among family and friends of the developer, Josh Wardle. Wordle’s popularity soared, reaching 3 million users after The New York Times purchased the game in January 2022. Today, some 2 million play Wordle daily. It is recreated in 50 languages globally.

Dilger’s suspicions arose while studying the game’s statistics published daily by The Times.

“I noticed one day an awful lot of people answered with one guess and thought, ‘that’s strange,'” Dilger said. “And then I paid attention to it and it was happening day after day. Well, I’m a science nerd and wanted to know what’s going on.”

Dilger imported statistics covering four months of user guesses into an Excel spreadsheet. His report, “Wordle: A Microcosm of Life. Luck, Skill, Cheating, Loyalty, and Influence!” appeared in the preprint server arXiv Sept. 6.

The game has a data bank containing 2,315 words, good for five years of play. (There actually are more than 12,000 five-letter words in the English language, but The Times weeded out the most obscure ones.)

Dilger calculated that the odds of randomly guessing the day’s word at 0.043%, totaling 860 players. Yet, Times statistics show that the number of players making correct first guesses in each game never dipped below 4,000.

“Do I mean to tell you that never, not once, was the share percent of the first guess less than 0.2%? Yup!” Dilger asserted.

He went further. His numbers are based on the 2,315-word master list compiled by The Times, but 800 of those words have already been used. Most players are not likely to know that detail, but if they did, and they excluded words already played, their odds of guessing the correct word would rise slightly. Yet, according to Dilger, their odds would still be a low 0.066%.

“Yet, it happens consistently every day,” Dilger said. “Some days it’s as high as 0.5%,” which would be 10,000 players.

He also noted how unlikely it would be that a user would correctly guess such poor first-choice candidates as “nanny” and “igloo.” Players gain maximum advantage when they surmise words with non-repeating characters and as many vowels as possible. “Nanny” repeats one letter three times and uses only two vowels. “Igloo” not only is a relatively rare word, but contains only two vowels, repeating one of them.

“What shall we call these people?” He asked. “‘Cheaters’ comes to mind, so that’s what I call ’em.”

Dilger did not offer any explanation for such nefarious behaviour, other than to say that many players “became frustrated at some point in the game and then felt joy or relief after having surpassed the hurdle with a cheat.”

“We are baffled as to how first-word cheaters actually have fun playing,” Dinger said, “but that does not diminish our enjoyment of the game.”

He might have quoted former wrestler, actor, philosopher and governor of Minnesota Jesse Ventura, who once suggested, “Winners never cheat, and cheaters never win.” Except maybe in Wordle.

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Credit of the article given to Peter Grad, Phys.org

You ordered a pizza for your party, but the restaurant forgot to slice it – these mathematical tricks can help you cut it evenly, says Katie Steckles.

Fairness is important – in life, and in pizza. If you want to cut a pizza into equal-sized pieces, the difficulty will depend on how many people you need to share it between. Luckily, mathematics has some tricks to keep things equal.

For example, if the number of people you are sharing a pizza between is a power of two – one, two, four, eight, 16 – cutting the pizza into as many slices is easy. For one piece, obviously no cuts are needed. For each larger power of two, a cut across through the centre of the pizza – cutting all of the existing pieces exactly in half – will result in pieces of equal size.

Some numbers will be much harder: prime numbers, by definition, can’t be divided easily. Luckily, geometry can help.

If you need to cut a pizza into five equal pieces, first grab a long, thin, rectangular strip of paper. Tie the paper in an ordinary overhand knot, like you would tie in a piece of string. Then, keeping the ends flat, pull gently to tighten the knot. The whole thing will flatten and come together – stop pulling when you can’t go any further without it wrinkling.

The flat shape you are looking at should now be vaguely familiar, if you ignore the two ends of paper sticking out. Fold these ends into the middle, or cut them off, and you will have a shape with five straight edges, created purely by the shape of the knot. Yes, that is right – you have made a perfect regular pentagon, with five equal-length sides and five equal angles at the corners.

It is possible to prove this mathematically by showing that all the folds you make in the paper strip are at 72 degrees to the parallel edges of the strip. But for simplicity, because the paper is the same width everywhere, and weaves in and out five times in the right way, these will be five equal edges. And more importantly, the pentagon’s corners are equally spread around a circle – making it the perfect guide for pizza slicing.

Place your pentagon in the centre of the pizza, then cut along lines radiating out from the centre of the pentagon and through each corner. And presto: you have a pentagonal pizza party for five. This paper-strip method can be used whenever you are in a pentagon-based emergency.

You can use the same technique to produce a shape with any odd number of sides by creating a more complex knot with the strip passing through the middle more times, although the strip of paper needs to be increasingly thin and it takes a lot more patience to pull the ends through and carefully flatten out the shape.

Combined with our existing halving methods, you can now produce any number of slices you like. The same results can be extended to any other round food – thanks to maths, the world is your cheesecake.

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*Credit for article given to Katie Steckles *

A major new study has revealed that American teenagers are more likely than any other nationality to brag about their math ability.

Research using data from 40,000 15-year-olds from nine English-speaking nations internationally found those in North America were the most likely to exaggerate their mathematical knowledge, while those in Ireland and Scotland were least likely to do so.

The study, published in Assessment in Education: Principles, Policy & Practice, used responses from the OECD Programme for International Student Assessment (PISA), in which participants took a two-hour math test alongside a 30-minute background questionnaire.

They were asked how familiar they were with each of 16 mathematical terms—but three of the terms were fake.

Further questions revealed those who claimed familiarity with non-existent mathematical concepts were also more likely to display overconfidence in their academic prowess, problem-solving skills and perseverance.

For instance, they claimed higher levels of competence in calculating a discount on a television and in finding their way to a destination. Two thirds of those most likely to overestimate their mathematical ability were confident they could work out the petrol consumption of a car, compared to just 40% of those least likely to do so.

Those likely to over-claim were also more likely to say if their mobile phone stopped sending texts they would consult a manual (41% versus 30%) while those less likely to do so tended to say they would react by pressing all the buttons (56% versus 49%).

Over-claimers were also more likely to say they were popular with their peers at school, although the evidence was less strong on this topic.

Overall, boys were more likely to overclaim than girls, and those from advantaged backgrounds were more likely to do so than those from less advantaged groups. In most countries, immigrants were more likely to do this than the native-born, particularly in Northern Ireland and New Zealand although not in the United States.

Three broad clusters of countries emerged, with the United States and Canada at the top of the rankings when it came to excessive claims on math knowledge, and with Ireland, Northern Ireland and Scotland at the bottom. In the middle were Australia, New Zealand, England and Wales.

The report’s lead author is John Jerrim, Professor of Education and Social Statistics at the UCL Institute of Education. “Our research provides important new insight into how those who over-claim about their math ability also exhibit high levels of over-confidence in other areas,” he said.

“Although ‘overclaiming’ may at first seem to be a negative social trait, we have previously found that overconfident individuals are more likely to land top-jobs. The fact that young men tend to overclaim their knowledge more than young women, and the rich are more likely to overclaim than the poor, could be related to the different labor market outcomes of these groups.”

Students were shown a list of 16 items and asked to indicate their knowledge of each on a five-point scale ranging from “never heard of it” to “know it well, understand the concept.” They were:

1. Exponential function
2. Divisor
3. Quadratic function
4. Proper number
5. Linear equation
6. Vectors
7. Complex number
8. Rational number
9. Radicals
10. Subjunctive scaling
11. Polygon
12. Declarative fraction
13. Congruent figure
14. Cosine
15. Arithmetic mean
16. Probability

Numbers 4, 10 and 12 were fake terms.

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Credit of the article given to Taylor & Francis

Richard Schwartz, a mathematician at Brown University has found a solution to the problem of how small a Möbius band can be made without intersecting itself—at least for a smooth piece of paper. The paper is published on the arXiv preprint server.

A Möbius strip (or band) is both a physical and mathematical object. A sample can be constructed by twisting a simple strip of paper one time and then taping the ends together. Since they were first discovered back in the mid-1800s, mathematicians have been scratching their heads trying to determine one simple constraint—what is the shortest strip necessary for making one? Back in the late 1970s, a pair of mathematicians, Charles Sidney Weaver and Benjamin Rigler Halpern, found that the problem could be made simpler by allowing self-intersections—that changed the problem to one that involved seeking the minimum amount of strip needed to avoid self-intersections.

Four years ago, Schwartz found himself intrigued by the problem and, as he describes in his paper, became “hooked” on finding a solution. Two years ago, he thought he had finally found it and published a proof showing his work—it involved breaking down the problem into multiple pieces and then using geometry principles to solve the puzzle as a whole.

Unfortunately, there turned out to be a major flaw in his work that he did not discover until much more recently. He found it by creating physical samples and cutting them in different ways to see how they worked on a deeper level. He discovered that the 2D strip was not shaped like a parallelogram as had been thought—instead, it was a trapezoid.

Inspired by his discovery, he went back to this original proof and corrected the error, and in doing so, found that the proof worked much better than it had originally. It was also much simpler. He also used the proof to work out the optimization problem, and got what he was hoping for: √3. He notes that while pleased with his own work, he has already turned his attention to another problem—to determine how short a band can be if it is twisted three times instead of once.\

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Credit of the article given to Bob Yirka, Phys.org