mathematics

Here’s a famous teaser that I don’t really like much: What number comes next in the sequence 8, 5, 4, 9, 1, 7,…? It’s not so obvious. It becomes a little easier when we write the sequence in words: eight, five, four, nine, one, seven, …. Spot it now? The answer is 6, because the digits are given in alphabetical order (and six comes next). It’s a fine trick, but it preys on the forgivable assumption that a pattern in a sequence of numbers will have something to do with numbers. I prefer when the parameters of a puzzle are…

Despite the wild success of ChatGPT and other large language models, the artificial neural networks (ANNs) that underpin these systems might be on the wrong track.For one, ANNs are “super power-hungry,” said Cornelia Fermüller, a computer scientist at the University of Maryland. “And the other issue is lack of transparency.” Such systems are so complicated that no one truly understands what they’re doing, or why they work so well. This, in turn, makes it almost impossible to get them to reason by analogy, which is what…

“In the old days, we only got oranges as presents—and we were happy about it!” This is a phrase you sometimes hear when an older person criticizes the lavish masses of gifts today’s children receive. What they rarely mention is the gift wrapping. Let’s say you wanted to give five oranges as a gift: How would you arrange the fruits so that they consumed as little space and wrapping paper as possible? As it turns out, there’s a lot of math behind this seemingly innocuous question. After all, it took more than 400 years to…

In the fall of 2017, Mehtaab Sawhney, then an undergraduate at the Massachusetts Institute of Technology, joined a graduate reading group that set out to study a single paper over a semester. But by the semester’s end, Sawhney recalls, they decided to move on, flummoxed by the proof’s complexity. “It was really amazing,” he said. “It just seemed completely out there.”The paper was by Peter Keevash of the University of Oxford. Its subject: mathematical objects called designs.The study of designs can be traced back to 1850,…

Three years ago Christopher Havens, who has been serving a prison sentence of more than two decades for murder, published a discovery in number theory from his cell. A significant class of fractions, he and three co-authors showed, often maintains a regular structure after being transformed algebraically. Havens’s achievement was singular in another respect: he did not have access to computers, which mathematicians commonly program to tackle aspects of such calculations, so he painstakingly pieced his research together by…

ChatGPT is an amazing tool, but plug-ins make it even more so by unlocking a range of exciting new abilities. From booking a restaurant table for you to custom designing t-shirts based on your prompts, ChatGPT plug-ins are the future of AI chatbots. Until the next big thing comes along, at least. Here are some of the best ChatGPT plug-ins you can use to leverage AI in ways you never even dreamed of. How to use ChatGPT plug-ins In order to run ChatGPT with plug-ins enabled, you need to be a ChatGPT Plus subscriber.…

When I tell people I’m a mathematician, one of the first things many of them say is “I’m really bad at maths”. Sometimes I can detect a touch of pride in their voices, but I suspect this is more like bravado – an attempt to hide a certain amount of insecurity, or block out often painful classroom memories.I’m sad that the subject is so polarising, with some believing they are “maths people” and others convinced they’re bad at it. But very few aspects of human behaviour are so black and white. Our abilities might fall…

Solving GATE Previous Year’s Questions (PYQs) not only clears the concepts but also helps to gain flexibility, speed, accuracy, and understanding of the level of questions generally asked in the GATE exam, and that eventually helps you to gain good marks in the examination. Previous Year Questions help a candidate practice and revise for GATE, which helps crack GATE with a good score. Discrete Mathematics Previous Year GATE Questions help in analyzing the question pattern of a subject and marking scheme as well as helps…

Solving GATE Previous Year’s Questions (PYQs) not only clears the concepts but also helps to gain flexibility, speed, accuracy, and understanding of the level of questions generally asked in the GATE exam, and that eventually helps you to gain good marks in the examination. Previous Year Questions help a candidate practice and revise for GATE, which helps crack GATE with a good score. Engineering Mathematics Previous Year GATE Questions help in analyzing the question pattern of a subject and marking scheme as well as it…

Then last fall, Milman came up for sabbatical and decided to visit Neeman so the pair could make a concentrated push on the bubble problem. “During sabbatical it’s a good time to try high-risk, high-gain types of things,” Milman said.For the first few months, they got nowhere. Finally, they decided to give themselves a slightly easier task than Sullivan’s full conjecture. If you give your bubbles one extra dimension of breathing room, you get a bonus: The best bubble cluster will have mirror symmetry across a central…