All posts tagged: maths

The maths meme that has been distracting mathematicians for a century

The maths meme that has been distracting mathematicians for a century

Published by skeptic

A tree-like form arises out of numerical connections in a famous maths puzzle known as the Collatz conjecture Marzio De Biasi/Algoritmarte Almost a century ago, a mathematician came up with a puzzle that was so seemingly simple and yet so fiendishly difficult that it has been distracting other mathematicians ever since. It has become a meme that jumps from brain to brain, with many people claiming to have solved it, only to have their hopes dashed as the proof unravels. And be warned – once I explain the rules, you will immediately want to start playing around with it yourself, and I take no responsibility for how much of your time you waste. It starts a bit like a magic trick. Pick a number, any number – well, at least any positive whole number; don’t try to get clever with something like pi. If it is an even number, divide it by 2. If it is an odd number, multiply it by 3 and add 1. Next, apply the same rules to the resulting number. …

A golden age of maths is dawning and mathematicians are freaking out

A golden age of maths is dawning and mathematicians are freaking out

Published by skeptic

I am attempting to solve a mathematical conundrum that has stumped many of humanity’s greatest thinkers. I have zero mathematical training, apart from a distant undergraduate physics degree, which should put my odds of success at slim to none. But I also have a trick up my sleeve – a kind of mathematical genie that can conjure arcane secrets seemingly out of thin air. I make a short request concerning an esoteric conjecture in number theory, then cross my fingers. Perhaps “genie” is a bit too strong – I’m simply using GPT 5.5 Pro, the latest iteration of OpenAI’s flagship model. But for mathematicians, modern AI models appear to have a spark of magic. Even in an era of rapid progress, the growth in AI’s mathematical ability is stunning. In just a few months, many prominent mathematicians have walked back previous scepticism and replaced it with sweeping predictions, whispering behind closed doors about job concerns and whether it is even worth embarking on a particular research project if AI might get there first. In April, …

The mathematician who doesn’t exist

The mathematician who doesn’t exist

Published by skeptic

A secret society of mathematicians has been working under a pseudonym for nearly a century Shutterstock/Stephen Ray Chapman One of the most important mathematicians in the world has been working for almost a century, producing dozens of books totalling thousands of pages that have served as a guiding light for the entire field. His name is Nicolas Bourbaki, and he doesn’t exist. Bourbaki is a pseudonym for a secret society of mathematicians. First formed in France in 1934, the group began with a simple goal: to update mathematical textbooks and make them more suitable for a contemporary audience. Instead, it created an entirely new way of writing mathematics that would make waves for decades. Initially, the group thought that its work would be around a thousand pages long and take six months. By 1935, Bourbaki had decided to write a series of six books, each building upon the previous one to “provide a solid foundation for the whole body of modern mathematics”, as later stated in an explanatory introduction. The group thought it would run …

Exam aids for maths and science GCSE exams to continue

Exam aids for maths and science GCSE exams to continue

Published by skeptic

Pupils will be continue to be given formulae and equation sheets for some GCSE subjects ahead of curriculum reforms, Ofqual has confirmed. The exams regulator decided exam boards will still be required to provide the exam aids for maths, physics and combined science, up to and including 2030 and 2031, the remaining lifetime of these qualifications. They had previously been confirmed until 2028. The aids were first introduced in 2022 due to concerns about the impact of the pandemic on learning. The requirement has been repeatedly extended. The government said in its response to the curriculum and assessment review that it would consider whether students should be required to memorise and recall each formula and equation, in reformed subjects from 2029 or 2030. Earlier this year, schools minister Georgia Gould said for the lifetime of these existing subjects pupils would not be required to memorise these for assessment purposes. Ofqual launched a three-week consultation and received more than 2,100 responses. Of these, 91.7 per cent supporting the proposals for a formulae sheet and 93 per cent supporting …

Maths pupils ‘make slower progress in mixed ability classes’

Maths pupils ‘make slower progress in mixed ability classes’

Published by skeptic

Pupils in mixed-ability classes make slower progress in maths compared to those grouped by attainment, a new report suggests. Using sets for maths also did not “significantly harm” the attainment of students from poorer backgrounds, or those with low attainment earlier in school. The report was published by the Education Endowment Foundation today, based on research conducted by the UCL Institute of Education. It investigated the impact of different ways of putting pupils in maths classes. The study compared the attainment and self-confidence of year 7 and 8 pupils taught in mixed-ability classes with those taught in sets between September 2022 and July 2024. Of the 97 schools that took part, 28 of them used mixed attainment groups while 69 used setting for maths. It found pupils in schools with mixed-ability classes made one month’s less progress in maths compared to pupils who were grouped based on attainment. Pupils with higher previous attainment in maths made about two months’ less progress when put in mixed ability classes rather than sets. But pupils with lower previous …

Greatest science books: Fermat’s Last Theorem is still a must-read about a 350-year maths secret

Greatest science books: Fermat’s Last Theorem is still a must-read about a 350-year maths secret

Published by skeptic

How does Simon Singh’s classic popular science book Fermat’s Last Theorem hold up today? Did you know the number 26 is rather special? It is the only number that sits directly between a square number (25 or 52) and a cube number (27 or 33). And to be clear, it’s not merely that we’ve never found another case of this square-cube sandwich. We know for certain that there isn’t another one between zero and infinity. Simon Singh’s 1997 book Fermat’s Last Theorem is an exploration of mathematical proof – what it means, how it’s obtained, and what drives those who so passionately seek it. It tells the story of the quest for one particularly beguiling proof, which makes it a compelling read. But given that this proof took 350 years to emerge, it also ends up being a wonderful history of mathematics. For many of us, the meat of mathematics lies in a realm of abstract reasoning far beyond us. But for me, what makes this book an absolute treasure, even almost 30 years after …

ABC conjecture: The secret project to settle controversial maths proof with a computer

ABC conjecture: The secret project to settle controversial maths proof with a computer

Published by skeptic

In 2012, Shinichi Mochizuki published a paper claiming to provide a proof for the ABC conjecture in number theory Newscom/Alamy One of the most bitterly contested proofs in modern mathematics may be on the verge of being untangled. Two projects, both aiming to use a computer program to cast new light on the controversy, are now up and running – with one having operated in secret for more than two years already. The developments are a positive sign that the row might find a solution, say mathematicians. The saga began in 2012 when Shinichi Mochizuki at Kyoto University, Japan, claimed to have proved a famous idea called the ABC conjecture, posting a 500-page proof online. The conjecture is simple to state, concerning prime numbers involved in solutions to the equation a + b = c and how these numbers relate to each other. But solving it requires deep insights into the nature of how addition and multiplication interact. The answer also has far-reaching implications for other mathematical disciplines. Mochizuki’s proof was a mathematical bombshell, but …

Gödel’s incompleteness theorem: The man who ruined mathematics

Gödel’s incompleteness theorem: The man who ruined mathematics

Published by skeptic

Logician, mathematician, philosopher and destroyer Kurt Gödel Pictorial Press/Alamy Kurt Gödel, the man who ruined mathematics, was one of the most important thinkers of the 20th century. He was born in 1906, smack-bang in the middle of the greatest crisis that maths has ever known. Just a few decades later, he would help resolve this turmoil, but in doing so doom mathematicians to a smaller world than the one that came before. Mathematics, as an intellectual framework, is incredibly powerful. The entire point is taking one set of logical ideas and using them to build another, making maths the closest thing we have to a cognitive perpetual-motion machine – there is always a new mathematical idea lurking across the horizon, and we just need to assemble the steps to get there. Or so it might seem. But in reality, there is a dark fundamental truth at the heart of mathematics that places limits on our intellectual exploration. It is called Gödel’s incompleteness theorem. The story of this theorem begins in the late 19th century, when …

Mathematician Gerd Faltings wins 2026 Abel prize for solving 60-year-old mystery

Mathematician Gerd Faltings wins 2026 Abel prize for solving 60-year-old mystery

Published by skeptic

Gerd Faltings has won the 2026 Abel Prize Peter Badge/Typos1 Gerd Faltings has won the 2026 Abel Prize, considered the Nobel prize of mathematics, for a groundbreaking proof which took mathematics by storm in 1983. His contributions helped establish one of the most important fields in modern mathematics, arithmetic geometry. The crowning achievement of Faltings, who also won the prestigious Fields medal in 1986 for the same work, was proving the Mordell conjecture, a longstanding theorem first proposed by the Louis Mordell in 1922 which argues that increasingly complicated equations produce fewer solutions. Faltings, who is based at the Max Planck Institute for Mathematics in Germany, says he was “honoured” when he found out the news, but was reserved about the impact of his achievements. “Somebody said, about climbing Mount Everest, it’s because it’s there and it was a problem,” says Faltings. “I solved [the Mordell conjecture], but in the end it doesn’t allow us to cure cancer or Alzheimer’s, it’s just extending our knowledge of things.” The Mordell conjecture concerns Diophantine equations, a vast …