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Why this matters The area of a triangle is base times height over two; the area of a circle is the radius squared times pi. For familiar shapes like these, you just memorize a formula. But the bumpy area carved out by a curve winding this way and that has no memorized formula at all. So how do you measure the area under a curve? The answer people found long ago was surprisingly humble — if you cannot measure it, fill it with something you can. Pack the space under the curve densely with many very thin rectangles, and add up all their areas. A rectangle's area is width times height, which you can always measure. ...

Watch almost any promotional clip for a humanoid robot and one shot keeps turning up: a neat stack of folded laundry. LG Electronics showed off the trick early this year, and a run of home-humanoid makers soon followed — Figure AI, Sunday Robotics, 1X, Weave Robotics. Tesla got there first, releasing footage of its Optimus robot straightening a shirt back in January 2024. It looks a little odd, spending tens of millions of dollars to build a robot and then showing it off by folding shirts and trousers. There's a reason, though. What comes easily to human hands is brutal for a machine. A piece of laundry is crumpled differently every time, its front and back jumbled, and its shape shifts the instant you touch it. ...

Meta is reshuffling how it builds artificial intelligence once more, and this round the pitch is openness. The company says its large language models will now center on open weights, and it paired that message with something concrete: a freely available model named Muse Glimmer, plus a promise to publish the weights for its stronger system, Muse Spark 1.2, over the coming weeks. Chief executive Mark Zuckerberg wrapped the shift in an essay that runs past 6,000 words, setting out how he believes AI systems should be built and governed from here. Part of the piece reads as a jab at rivals. OpenAI and Anthropic keep their models proprietary, and both have lobbied Washington for cover against two things they treat as threats: large-scale distillation, in which one model trains another, and the open-weight systems pouring out of Chinese labs. Meta is choosing the other side of that line. ...

Why this matters An ordinary equation ends in a few numbers. But the answer to a differential equation is a function, not a number — a whole curve. And that answer always comes with a mysterious constant attached. $$\frac{dy}{dx} = 2x \;\Rightarrow\; y = x^{2} + C$$Here \(\dfrac{dy}{dx}\) is a single chunk of a symbol meaning "how fast \(y\) changes when \(x\) changes by a tiny amount," that is, the slope of the curve. And \(C\) is an arbitrary constant that could be \(3\), or \(-7\), or any value at all. ...

For years the worry in classrooms was plagiarism. Now it's almost the reverse: proving that a person, not a machine, did the writing. Schools and publishers have reached for a new class of software — AI detectors — that promise to spot text made by tools like ChatGPT. The catch is that these programs are shaky, their own makers admit it, and people keep leaning on them anyway. Start with where this came from. Anti-plagiarism software predates ChatGPT by a long stretch. Programs such as Turnitin check a submission against a huge library of web pages, journal articles and past work, hunting for sentences and phrases that line up too neatly. Turnitin even hands back a percentage meant to show how much of a student's text matches other sources. That figure was always slippery — a high score could mean deliberate copying or an innocent coincidence — and the false alarms pushed some teachers to quietly drop it. ...

For the first time, more than 55 million people open KakaoTalk in a given month. Figures Kakao released on the 10th put the app's monthly active users at 55.346 million in the second quarter. It first crossed the 50-million line back in the first quarter of 2018, which means stacking the next five million on top took nearly eight years. What draws the eye isn't the milestone itself but where the extra users came from. Domestic numbers kept climbing, yet this quarter carried a new factor from beyond the border: overseas, in Russia especially, KakaoTalk got tapped as a stand-in for Telegram. ...

Why this matters An ordinary equation usually answers with a number. "What number doubles to 4?" — the answer is just \(2\). "What number squares to 4?" — \(2\) and \(-2\), two of them. However many answers there are, they are a few points on the number line. A differential equation, though, answers differently. A differential equation is an equation with a derivative (an instantaneous rate of change) in it. For example, ...

When the internet stutters at home, most of us reach over and power-cycle the router. The technology LG Uplus said it demonstrated on the 9th hands that chore to the machine instead. Before anyone notices a slowdown, AI inside the router reads the situation and quietly sets it right. The core idea is self-healing. By the company's account, the demonstration confirmed the feasibility of an intelligent home network in which AI spots warning signs and responds while a customer is still one step away from feeling any drop in quality. The work was carried out on Wi-Fi 8, the next-generation wireless standard. ...

Pull an old Game Boy off the shelf and the first thing it demands is batteries — a lot of them. The handhelds that ruled portable play in the late '80s and '90s were designed around disposable cells, and they empty them fast. The good news is that the fix isn't exotic. Modern rechargeables have quietly taken most of the pain out of it. The advice here centers on the era's best-known battery-powered handhelds, the ones sold from the late 1980s into the first years of the 2000s. That covers Nintendo's Game Boy in its original, Color, Pocket and Advance versions, along with the Game Gear from Sega and the Lynx from Atari. Every one of them shipped before rechargeable cells became the habit most people keep now, which is exactly why they feel so thirsty today. ...

Why this matters Integrating a fraction often feels like a dead end. Meet an integral like this and you have no idea where to grab it. $$\int \frac{3x+1}{(x+1)(x-1)} \, dx$$Here \(\int\) is the long S-shaped integral sign, meaning "add up all the finely sliced pieces," and \(dx\) marks that the direction of that adding is \(x\). The denominator is tangled into a product \((x+1)(x-1)\), so as it stands not a single formula you know applies. ...