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The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery. Nichomauss' Theorem: the sum of the first cubes can be written as the square of the sum of the first integers, a statement that can be written as . Here, we use the same re-arrangement as the first proof on this page (the sum of first odd integers is a square). Here's another re-arrangement to see this: This also suggests the following alternative proof:
So you've volunteered to be a Firebreaker, and now you're staring down a paranatural threat with only a glorified super soaker in your hands. It happens to the best of us. FBC: Firebreak is developer Remedy Entertainment's first crack at a live-service multiplayer game. It's a Left 4 Dead-like horde shooter set in the Oldest House (from Control), an ever-shifting building that contains the world's foremost paranatural threats. Taking place six years after Remedy's 2019 game Control, FBC: Fireb
AI companies are battling to dominate the industry, but sometimes they’re also battling in Pokémon gyms. As Google and Anthropic both study how their latest AI models navigate early Pokémon games, the results can be as amusing as they are enlightening — and this time, Google DeepMind has written in a report that Gemini 2.5 Pro resorts to panic when its Pokémon are close to death. This can cause the AI’s performance to experience “qualitatively observable degradation in the model’s reasoning cap
All 17 experts ousted from the federal vaccine advisory committee have spoken out about the drastic changes that anti-vaccine advocate and current US health secretary Robert F. Kennedy Jr. has made since taking office. Those changes include unilaterally restricting access to COVID-19 vaccines and summarily firing the entire Advisory Committee on Immunization Practices (ACIP), which had guided federal vaccine policies for more than 60 years. "We are deeply concerned that these destabilizing deci
In context: The use of GenAI in the workplace is evolving at a breakneck pace. Two new reports – one from Gallup and another from Salesforce AI Research – paint a picture of both growing enthusiasm and persistent challenges as organizations and their employees adapt to this technological shift. Together, these studies reveal a central theme: while AI is being embraced more widely than ever, the gap between its promise and practical performance remains a significant hurdle for businesses and work
is a senior policy reporter at The Verge, covering the intersection of Silicon Valley and Capitol Hill. She spent 5 years covering tech policy at CNBC, writing about antitrust, privacy, and content moderation reform. The Senate confirmed Republican Olivia Trusty to serve on the Federal Communications Commission (FCC) on Tuesday, installing another nominee by President Donald Trump and ending the brief lack of quorum at the agency. The vote was 53-45, with Sen. John Fetterman (D-PA) being the on
Join the event trusted by enterprise leaders for nearly two decades. VB Transform brings together the people building real enterprise AI strategy. Learn more Google moved decisively to strengthen its position in the artificial intelligence arms race Monday, declaring its most powerful Gemini 2.5 models ready for enterprise production while unveiling a new ultra-efficient variant designed to undercut competitors on cost and speed. The Alphabet subsidiary promoted two of its flagship AI models—G
TL;DR Gemini 2.5 Flash-Lite is rolling out in preview. The stable versions of Gemini 2.5 Flash and Pro are also rolling out. The cost of input for Flash has gone up by $0.15, but the output has been reduced by $1.00. Google introduced Gemini 2.5 earlier this year, with Pro (experimental) being the first model in the series. A couple of months later, the company rolled out a faster model in early access, known as Gemini 2.5 Flash. Now the firm has an update on these two models and is launching
Amazon CEO Andy Jassy speaks during a keynote address at AWS re:Invent 2024, a conference hosted by Amazon Web Services, at The Venetian Las Vegas on December 3, 2024 in Las Vegas, Nevada. Amazon CEO Andy Jassy said Tuesday that the company's corporate workforce will shrink in the coming years as it adopts more generative artificial intelligence tools and agents. "We will need fewer people doing some of the jobs that are being done today, and more people doing other types of jobs," Jassy said
We designed Gemini 2.5 to be a family of hybrid reasoning models that provide amazing performance, while also being at the Pareto Frontier of cost and speed. Today, we’re taking the next step with our 2.5 Pro and Flash models by releasing them as stable and generally available. And we’re bringing you 2.5 Flash-Lite in preview — our most cost-efficient and fastest 2.5 model yet. Making 2.5 Flash and 2.5 Pro generally available Thanks to all of your feedback, today we’re releasing stable version
Foundation season three will pick up 152 years after the end of season two, which means a new set of supporting characters is on the way. However, given Foundation’s fondness for clones (be they organic or digital) and robots—as well as people who endure extended cryosleep—all the main faces will also be back. That time jump will advance the action to a time of great unrest, as predicted in season two by Gaal Dornick (Lou Llobell) using a combination of mathematics, the “prime radiant” quantum
Amazon CEO Andy Jassy speaks during a keynote address at AWS re:Invent 2024, a conference hosted by Amazon Web Services, at The Venetian Las Vegas on December 3, 2024 in Las Vegas, Nevada. Amazon CEO Andy Jassy said Tuesday that the company's corporate workforce will shrink in the coming years as it adopts more generative artificial intelligence tools and agents. "We will need fewer people doing some of the jobs that are being done today, and more people doing other types of jobs," Jassy said
Google has announced a big expansion of its Gemini AI model family today. After months of tweaking and tuning, the high-power Gemini 2.5 Pro is leaving preview and is ready for developers to build on. Meanwhile, Google is offering a peek at its upcoming high-efficiency model, known as Gemini 2.5 Pro Flash-Lite. Try as it might, Google can't get away from confusing model names. Google's AI aspirations have been looking up in 2025 with the debut of Gemini 2.5. These models showed a marked improve
Microsoft’s Game Pass is having a very impressive 2025 where the quality and cadence of releases is concerned, and things are showing no signs of slowing down as we settle into the summer months. Highlights of the June lineup include co-op FPS FBC: Firebreak, which joins the service today, Sloclap’s highly-anticipated soccer game, Rematch, and a trio of Warcraft remasters. Let’s start with probably the most significant addition. Back in May, Engadget’s Jessica Conditt got to play Remedy’s FBC:
We designed Gemini 2.5 to be a family of hybrid reasoning models that provide amazing performance, while also being at the Pareto Frontier of cost and speed. Today, we’re taking the next step with our 2.5 Pro and Flash models by releasing them as stable and generally available. And we’re bringing you 2.5 Flash-Lite in preview — our most cost-efficient and fastest 2.5 model yet. Making 2.5 Flash and 2.5 Pro generally available Thanks to all of your feedback, today we’re releasing stable version
A hot potato: Now that people are no longer angry over Borderlands 4 possibly costing $80 (it's actually $70 for the standard edition), something else is causing plenty of outrage: the game's PC requirements. Asking for an RTX 3080 as the recommended GPU does seem excessive, and even the minimum specs are higher than expected. Borderlands 4's system requirements on its Steam page are certainly higher than one would expect. Even the bare minimum requirements, where we usually see ancient and/or
A legal expert found that Meta's AI is able to spit out entire portions of books verbatim — and if he's right, it could be seriously bad news for the company and its CEO Mark Zuckerberg. First, a quick primer. All the AI that's commercially buzzy at the moment, like OpenAI's ChatGPT or Meta's Llama, is trained by feeding in huge amounts of data. Then researchers do a bunch of number crunching using algorithms, basically teaching the system to recognize patterns in all that data so thoroughly th
Alaska got its first-ever heat advisory over the weekend, which is in effect until June 17. Over the last few days, temperatures in central Alaska have reached 86 degrees Fahrenheit (that's 30 Celsius). It's not as if Alaska never gets hot in the summer. However, as Jason Laney, a warning coordination meteorologist with the National Weather Service in Fairbanks, Alaska, told CNET, "The frequency with which the heat is coming up is starting to be a bit of a concern." This heat advisory seems pr
Last month, Suzanne Smith-Darley felt fantastic. She had just bought a used Chanel handbag from a Japanese seller on eBay for $800—a steal compared to the original asking price of $1,400. About a week later an email arrived that crushed her: DHL was demanding a $142 fee for US tariffs before it would deliver the well-worn medallion tote to Smith-Darley’s Atlanta doorstep. “It goes to Japan, has a whole life, and it could be in the trash literally,” she says. “I’m willing to pick it out of the tr
I'm in the midst of putting together a buying guide of indoor vertical gardening systems, and the Gardyn—the 30-plant Home 4.0, to be exact—was the first tester to arrive at my house. I had it unboxed and set up within a couple of hours, lights on and water pump running. I'm already a pro! I thought. Sure enough, within a couple of weeks, all of Gardyn's proprietary seed-filled yCubes had sprouted, and a couple of weeks after that, I was harvesting bowlfuls of herbs and salad greens. Even thoug
Why it matters: Preserving retro games and hardware becomes increasingly difficult as the years pass and essential components grow rarer. Reverse engineering allows modders and repair shops to create substitutes that function identically to the original parts. Now, a new project aims to bring that level of flexibility to the original PlayStation. Italian engineer Lorentio Brodesco recently became the first person to successfully reverse-engineer the original PlayStation's motherboard, creating
Edgar Cervantes / Android Authority TL;DR Google has recently updated Gemini with a boatload of in-development UI tweaks. Some include dropping the “Gemini” name from headers, and rebranding “Saved info.” We also see further progress on multiple changes to the way you view and access chat history. Update 2: June 17, 2025 (09:25 AM ET): The updated sidebar is now rolling out in beta with Google app v16.23.69. AssembleDebug / Android Authority However, the ability to search past conversation
Device management vendor Fleet has raised $27 million in Series B funding to accelerate development of its open device management platform. The round was led by Ten Eleven Ventures and brings Fleet’s total funding to $52.3 million. The company reports 6x revenue growth over the past two years and plans to use the funding to expand adoption across enterprises looking for flexible deployment options. Some of my favorite gear eufyCam 2C Upgrade your home security with wireless cameras that include
iOS 26 introduces a lot of big changes to how the iPhone software looks and feels, but there are plenty of updates for system apps too, including Notes, Maps, Photos, and more. Apple’s Reminders app gets a variety of new features too, here’s everything coming to Reminders in iOS 26. New Liquid Glass design Apple’s new ‘Liquid Glass’ design impacts all built-in apps in iOS 26, but certain apps feel more changed than others. When viewing your list of tasks, Reminders’ design has only subtle cha
You’d assume that the latest game from the studio that made Little Nightmares 2 is going to be a bit spooky and dark, and Section 9’s End of Abyss is that. However, it's also a little easier to describe and explain: an atmospheric sci-fi shooter with Metroidvania-style exploration. While there's no shortage of Metroidvanias to draw inspiration from, when I played a demo at SGF 2025, all I could think of was Alien. It might be the scanner, which the protagonist can use to glean information, spot
Email hosting provider Cock.li has confirmed it suffered a data breach after threat actors exploited flaws in its now-retired Roundcube webmail platform to steal over a million user records. The incident exposed all users who had logged in to the mail service since 2016, estimated at 1,023,800 people, along with contact entries for an additional 93,000 users. Cock.li is a Germany-based free email hosting provider with a privacy-focused ethos and lax moderation policies, run by a single operato
Maria Diaz/ZDNET Have you ever found yourself in a thermostat tug-of-war? One room feels like a sauna, another like an icebox, all while your trusty thermostat claims your home is at a perfectly average and comfortable 72 degrees? The truth is, you're probably not imagining things, and the culprit could be hanging right there on your wall. As it turns out, where your thermostat is located can make a world of difference to both your comfort and your energy bills. Also: How to disable ACR on yo
Google Wave's inbox, as seen on the Official Google Blog. ((Courtesy Google)) In May, Google announced Wave, the company's attempt to reinvent internet communication. Brothers Lars and Jens Rasmussen, the Australian creators of Google Maps, have been working on the project for the last two years. On Sept. 30, Google opened its preview of Wave beyond the 6,000 or so software developers who are currently helping to build the service. About 100,000 users can now see what Wave is all about. Google
If you’re out there building yourself your first gaming PC, let me just say it’s not as hard as it looks. Once you find all the parts you need for the budget you want to spend, it is just a matter of putting them all together. And there and plenty of YouTube videos that will show you exactly what to do if you are nervous on that front. No, the hard part is actually that first part — finding the parts your need for the budget you want to spend. There are a lot of options out there with hard-to-de
Today, AMD’s Infinity Fabric interconnect is ubiquitous across the company’s lineup. Infinity Fabric provides well-defined interfaces to a transport layer, and lets different functional blocks treat the interconnect as a black box. The system worked well enough to let AMD create integrated GPU products all the way from the Steam Deck’s tiny van Gogh APU, to giant systems packing four MI300A chips. Across all those offerings, Infinity Fabric enables coherent memory access as CPU and GPU requests