Author: IMC

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 *

Five secret phrases used to create the encryption algorithms that secure everything from online banking to email have been lost to history – but now cryptographers are offering a bounty to rediscover them.

Could you solve a cryptography mystery?

Secret phrases that lie at the heart of modern data encryption standards were accidentally forgotten decades ago – but now cryptographers are offering a cash bounty for anyone who can figure them out. While this won’t allow anyone to break these encryption methods, it could solve a long-standing puzzle in the history of cryptography.

“This thing is used everywhere, and it’s an interesting question; what’s the full story? Where did they come from?” says cryptographer Filippo Valsorda. “Let’s help the trust in this important tool of cryptography, and let’s fill out this page of history that got torn off.”

The tool in question is a set of widely-used encryption algorithms that rely on mathematical objects called elliptic curves. In theory, any of an infinite number of curves can be used in the algorithms, but in the late 1990s the US National Security Agency (NSA), which is devoted to protecting domestic communications and cracking foreign transmissions, chose five specific curves it recommended for use. These were then included in official US encryption standards laid down in 2000, which are still used worldwide today.

Exactly why the NSA chose these particular curves is unclear, with the agency saying only that they were chosen at random. This led some people to believe that the NSA had secretly selected curves that were weak in some way, allowing the agency to crack them. Although there is no evidence that the elliptic curves in use today have been cracked, the story persists.

In the intervening years, it has been confirmed that the curves were chosen by an NSA cryptographer named Jerry Solinas, who died earlier this year. Anonymous sources have suggested that Solinas chose the curves by transforming English phrases into a string of numbers, or hashes, that served as a parameter in the curves.

It is thought the phrases were along the lines of “Jerry deserves a raise”. But rumours suggest Solinas’s computer was replaced shortly after making the choice, and keeping no record of them, he couldn’t figure out the specific phrases that produced the hashes used in the curves. Turning a phrase into a hash is a one-way process, meaning that recovering them was impossible with the computing power available at the time.

Dustin Moody at the US National Institute of Standards and Technology, which sets US encryption standards, confirmed the stories to New Scientist: “I asked Jerry Solinas once, and he said he didn’t remember what they were. Jerry did seem to wish he remembered, as he could tell it would be useful for people to know exactly how the generation had gone. I think that when they were created, nobody [thought] that the provenance was a big deal.”

Now, Valsorda and other backers have offered a $12,288 bounty for cracking these five hashes – which will be tripled if the recipient chooses to donate it to charity. Half of the sum will go to the person who finds the first seed phrase, and the other half to whoever can find the remaining four.

Valsorda says that finding the hashes won’t weaken elliptic curve cryptography – because it is the nature of the curves that protects data, not the mathematical description of those curves – but that doing so will “help fill in a page of cryptographic history”. He believes that nobody in the 1990s considered that the phrases would be of interest in the future, and that the NSA couldn’t have released them anyway once they discovered that they were jokey phrases about one of their staff wanting a raise.

There are two main ways someone could claim the prize. The first is brute force – simply trying vast numbers of possible seeds, and checking the values created by hashing them against the known curves, which is more feasible than in the 1990s because of advances in computing power. 

But Valsorda says someone may already have the phrases written down. “Some of the people who did this work, or were in the same office as the people who did this work, probably are still around and remember some details,” he says. “The people who are involved in history sometimes don’t realise the importance of what they remember. But I’m not actually suggesting anybody, like, goes stalking NSA analysts.”

Keith Martin at Royal Holloway, University of London, says that the NSA itself would be best-equipped to crack the problem, but probably has other priorities, and anybody else will struggle to find the resources.

“I would be surprised if they’re successful,” he says. “But on the other hand, I can’t say for sure what hardware is out there and what hardware will be devoted to this problem. If someone does find the [phrases], what would be really interesting is how did they do it, rather than that they’ve done it.”

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*Credit for article given to Matthew Sparkes*

College students learn more calculus in an active learning course in which students solve problems during class than in a traditional lecture-based course. That’s according to a peer-reviewed study my colleagues and I published in science. We also found that college students better understood complex calculus concepts and earned better grades in the active learning course.

The findings held across racial and ethnic groups, genders and college majors, and for both first-time college and transfer students—thus, promoting success for all students. Students in the active learning course had an associated 11% higher pass rate.

If you apply that rate to the current 300,000students taking calculus each year in the U.S., it could mean an additional 33,000 pass their class.

Our experimental trial ran over three semesters—fall 2018 through fall 2019—and involved 811 undergraduate students at a public university that has been designated as a Hispanic-serving institution. The study evaluated the impact of an engagement-focused active learning calculus teaching method by randomly placing students into either a traditional lecture-based class or the active learning calculus class.

The active learning intervention promoted development of calculus understanding during class, with students working through exercises designed to build calculus knowledge and with faculty monitoring and guiding the process.

This differs from the lecture setting where students passively listen to the instructor and develop their understanding outside of class, often on their own.

An active learning approach allows students to work together to solve problems and explain ideas to each other. Active learning is about understanding the “why” behind a subject versus merely trying to memorize it.

Along the way, students experiment with their ideas, learn from their mistakes and ultimately make sense of calculus. In this way, they replicate the practices of mathematicians, including making and testing educated guesses, sense-making and explaining their reasoning to colleagues. Faculty are a critical part of the process. They guide the process through probing questions, demonstrating mathematical strategies, monitoring group progress and adapting pace and activities to foster student learning.

Florida International University made a short video to accompany a research paper on how active learning improves outcomes for calculus students.

Why it matters

Calculus is a foundational discipline for science, technology, engineering and mathematics, as it provides the skills for designing systems as well as for studying and predicting change.

But historically it’s been a barrier that has ended the opportunity for many students to achieve their goal of a STEM career. Only 40% of undergraduate students intending to earn a STEM degree complete their degree, and calculus plays a role in that loss. The reasons vary depending on the student. Failing calculus can be a final straw for some.

And it is particularly concerning for historically underrepresented groups. The odds of female students leaving a STEM major after calculus is 1.5 times higher than it is for men. And Hispanic and Black students have a 50% higher failure rate than white students in calculus. These losses deprive the individual students of STEM aspirations, career dreams and financial security. And it deprives society of their potentially innovative contributions to solving challenging problems, such as climate resilience, energy independence, infrastructure and more.

What still isn’t known

A vexing challenge in calculus instruction—and across the STEM disciplines—is broad adoption of active learning strategies that work. We started this research to provide compelling evidence to show that this model works and to drive further change. The next step is addressing the barriers, including lack of time, questions about effectiveness and institutional policies that don’t provide an incentive for faculty to bring active learning to their classrooms.

A crucial next step is improving the evidence-based instructional change strategies that will promote adoption of active learning instruction in the classroom.

What’s next

Our latest results are motivating our team to further delve into the underlying instructional strategies that drive student understanding in calculus. We’re also looking for opportunities to replicate the experiment at a variety of institutions, including high schools, which will provide more insight into how to expand adoption across the nation.

We hope that this paper increases the rate of change of all faculty adopting active learning in their classrooms.

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Credit of the article given to Laird Kramer, The Conversation

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