In 2025, the Busy Beaver Challenge Community offers an unprecedented glimpse into the cutting-edge realm of mathematics, where large numbers are poised to challenge the very foundations of logical reasoning.
This exploration centers on the next number in the “Busy Beaver” sequence, a collection of rapidly increasing values that arise from a fundamental query: How can we determine whether a computer program has the potential to run indefinitely?
To answer this, researchers draw upon the seminal work of mathematician Alan Turing, who demonstrated that any computer algorithm could be modeled using a simplified mechanism called a Turing machine. More intricate algorithms correspond to Turing machines with expanded instruction sets or a greater number of states.
Each Busy Beaver number, denoted as BB(n), denotes the longest execution time achievable for an n-state Turing machine. For instance, BB(1) equals 1 and BB(2) equals 6, indicating that doubling the complexity of the algorithm extends its runtime sixfold. This growth escalates rapidly; for example, BB(5) reaches an astounding 47,176,870.
In 2024, members of the Busy Beaver Challenge succeeded in determining the exact value of BB(5), culminating a 40-year study into every Turing machine comprising five states. Consequently, 2025 became a year dedicated to pursuing BB(6).
In July, a member known as mxdys identified the lower bound for BB(6), revealing that its value is not only significantly larger than BB(5) but also dwarfs the number of atoms in the universe.
Due to the impracticality of expressing all its digits, mathematicians utilize a notation system called tetration, which involves exponentiating numbers repetitively. For example, 2 raised to the power of 2 results in 4, which can similarly be expressed as 2 raised to the power of 4, yielding 16. BB(6) is at least as large as 2 raised to the power of 2 raised to the power of 9, forming a towering structure of repeated squares.
Discovering BB(6) transcends mere record-setting; it holds significant implications for the field of mathematics. Turing’s findings assert the existence of a Turing machine behavior that eludes prediction within a framework known as ZFC theory, which underpins contemporary mathematics.
Researchers have previously indicated that BB(643) defies ZFC theory, but the potential for this occurrence in a limited number of cases remains uncertain, positioning the Busy Beaver Challenge as a vital contributor to advancing our understanding.
As of July, there were 2,728 Turing machines with six states still awaiting analysis of their stopping behavior. By October, that number diminished to 1,618. “The community is currently very engaged,” comments computer scientist Tristan Stellin, who introduced the Busy Beaver Challenge in 2022.
Among the remaining machines lies the potential key to precisely determining BB(6). Any one of these could be a crucial unknown, possibly revealing substantial limitations of the ZFC framework and contemporary mathematics. In the coming year, math enthusiasts worldwide are poised to delve deeply into these complexities.
GReactor actors have always fascinated artists. I suppose edmund keene He looks wild and devilish, like George Clint's Sir Giles Overreach. Ellen Terry as Lady Macbeth in a green gown saved by John Singer Sargent; and Ruskin Spear's research on oil. Laurence Olivier As Macbeth, who suffers and suffers from guilt. For over a century, it has also been possible to record the voices of leading actors. But what happens when images and sounds are combined?
One answer can be found at an innovative new exhibition. Shakespeare portrait On display at red eight gallerynext to the Royal Exchange on Cornhill in the City of London. The show consists of 10 digital portraits of live actors accompanied by speeches from Shakespeare's plays. It is easiest to explain with an example. As I sat under a large framed statue of Ian McKellen and chatted with the exhibition's creative director, I said: Arsalan Sattari-HicksI noticed that Sir Ian's head was moving from time to time, that his gaze was changing slightly, and that his features expressed different emotions. At one point, I even heard him recite part of “All the world's a stage” from As You Like It with his unique virtuosity. Richard Brierley, the gallery's director, told me: “Usually the portrait is passive and you are active. In this case, the portrait is active and you are passive.”
There it is in the eyes… Portrait of Juliet Stevenson. Photo: Stage block
I would like to qualify this by saying that the changes in the sitter's movements are often so subtle, almost imperceptible, that the viewer can activate the speech by pressing a button. But the overall effect was amazing and I was intrigued by how it was done. I'm told that Sattari-Hicks and a small crew record the sitters in a studio using state-of-the-art cameras, then adjust the images through hours of post-production. The audio text was selected by the actors in collaboration with Shakespeare director Ron Daniels, who curated the final version. Considering the extensive experience of the actors, the process usually takes 1 hour and 20 minutes.
I was struck by the intimacy of the experience. The faces of the actors are shown in close-up, as if they suddenly appeared on the gallery wall. And the text is spoken in a way that further enhances the pervasive calm. Patrick Stewart quotes Henry V's St. Crispin's Day speech and strips away the rhetoric to show a man talking to other men on the eve of battle. He suggests that Crispin's holiday “will be remembered from this day until the end of the war.” Olivier famously let his lines soar skyward in his films, and he did so with understated conviction.
Impressive members… Juliet Stevenson, Derek Jacobi, Ian McKellen, Patrick Stewart, Harriet Walter, Charles Dance. Photo: Stage block
The big question is, what happens next? The ten portraits on display are unique collectibles that will be sold to individuals or organizations. But they're the work of StageBlock, a company co-founded by Sattari-Hicks and Francesco Pierangeli that has ambitious plans for the future. They are already planning to record the second volume of A Portrait of Shakespeare in the new year. They also dream of extending the idea of living portraits to other authors and other cultures, and funneling 10% of the funds raised from introductions and exhibitions into the performing arts.
Given the impressive roster of names at the first exhibition, which also includes David Suchet, Juliet Stevenson, Adrian Lester, Simon Callow, Charles Dance and Frances Barber, there's a good chance the profits will continue. there is. This is a bold and visionary new idea in the digital age, but we hope it doesn't inhibit the talent of individual portraitists. For example, I value the concept of Salvador Dali's paintings. Olivier as Richard III On the walls of the gallery are displayed captivating images of McKellen gazing attentively and seductively.
ohOn a sweltering summer day in London, I found myself working in the middle of snow-covered Yosemite National Park, surrounded by floating apps and browser windows. Later, I’d reminisce about holidays from years ago, staring out at windswept Oregon beaches, sitting in a speeder on Tatooine watching Rogue One in 3D, and spending the night with a guided meditation.
These are the sort of immersive experiences Apple’s latest, and most expensive, gadget offers, blending the real and virtual worlds, all controlled by your eyes and hands. The Vision Pro may resemble virtual reality headsets like Meta’s Quest series, but it aims to be something much more.
But with a prohibitive price tag of £3,500 (€3,999 / $3,499 / AU$5,999) that most buyers won’t even consider, this cutting-edge tech marvel is best thought of as a glimpse into the near future of computing.
You can use your Mac’s screen, keyboard, and trackpad streamed to a simulated 4K display, alongside other windows and apps. Photo: Martin Godwin/The Guardian
Put on the headset and you’re transported to a photorealistic exotic location, or use the Digital Crown to increase or decrease immersion, seamlessly blending reality and the virtual world. The real world is sent through the camera to a crystal-clear display and displayed as pass-through video — far better than the competition, and so clear that you can read on your phone without taking off the headset.
Your content appears in a floating window fixed in 3D space, as big or small as you like. Even if you walk by the window, your content stays where you were and is instantly visible when you return. Just see the content you want and select it with a pinch of your fingers. Type directly by “touch” on the hover keyboard or scroll through sites like a giant virtual iPad.
Step into an immersive experience and go one step further by walking with dinosaurs, exploring the solar system or flying along neon-lit highways in rhythm games.
Third-party apps offer a variety of mixed reality and immersive experiences. astronomy (Top left), Jetpack Joyride 2 (Top right), Disney+ (Bottom left) and Luna (Bottom right). Photo: Samuel Gibbs/The Guardian
Heavyweight Technology
Vision Pro is the pinnacle of headset tech: The same M2 chip found in the 2022 MacBook Air runs apps, and its R1 chip processes input from the headset’s 12 cameras, five sensors, and six microphones. Combined, this delivers a smooth experience in both the real and virtual worlds displayed on two high-resolution Micro-OLED displays in front of your eyes.
The exterior cameras and sensors create a map of the real world, including objects like furniture and walls, and track the user’s position and hand movements. The interior camera monitors eye movements to interact with buttons and objects, making sure what you’re looking at is clear. The headset also features “Optic ID,” an alternative to Face ID, to seamlessly unlock and authenticate payments by scanning your iris.
The experience is exponentially better than anything that came before, and at times, it’s magical.
The various parts of the Vision Pro attach to each other via magnets and quick-release clips, including two types of straps. Photo: Martin Godwin/The Guardian
All the technology packed into the Vision Pro creates one major problem: weight. At up to 650 grams, it’s heavier than Apple’s largest iPad Pro and competing headsets like the Quest 3, which weighs 515 grams. And that doesn’t include the 353 gram battery, which connects to the headset with a cord so you can put it in your pocket or keep it on the desk in front of you.
During the ordering process, your face is scanned with an iPhone and a custom fit is created from nearly 200 combinations of strap sizes, “light seals” and cushioning, making it more comfortable than any other headset and leaving no goggle marks on your face.
What remains is the strain on my neck. After wearing the headset daily for a month, I can now manage sessions up to about two hours long. However, I still feel like I’ve given my neck a workout, and wearing it for long periods without taking proper breaks causes the same neck, shoulder and back pain I get when I’m hunched over a laptop all day.
The battery lasts for about 2-3 hours, which is plenty long enough for you to easily charge it while sitting at your desk or on the couch, but this headset isn’t designed to be easily shared, and even if you manage to get a good fit on your guest’s face, you’ll need to redo the eye-tracking setup for five minutes to get it to work temporarily.
We’re only scratching the surface in productivity improvements
I placed my Mac display in the center with various windows around it, and some behind and above it, and the screenshots in the headset don’t do justice to how it will look in person. Photo: Samuel Gibbs/The Guardian
The Vision Pro is different from other headsets in that it’s fully integrated into the Apple ecosystem — more like a Mac than an iPhone — allowing you to create an entire app and productivity environment anywhere, without the need for multiple monitors.
It comes with many familiar apps, including Apple’s Mail, Messages, Notes, Keynote, Freeform, and Photos, and many others are available as “compatible” apps, including Microsoft’s Word and Excel, but it doesn’t include Google apps like Gmail or Drive, and only some of them work properly as web apps in Safari.
The headset tracks your hand movements relative to virtual objects, and here we see the common two-handed pinch-to-zoom gesture to increase the size of a photo. Photo: Martin Godwin/The Guardian
Using the Vision Pro as part of a productivity setup is great, but that’s only scratching the surface of what the headset can do.
Apps for Vision Pro are varied: some simply drag 2D experiences into the 3D space of the headset, like games played on a TV screen placed within the environment, while others are fully immersive environments you can walk around in.
The Apple TV app lets you enter a virtual cinema and choose your row and seat, while Disney+ lets you sit on a couch in Avengers Tower or the aforementioned speeder on Tatooine. On both services, the 3D movies look especially good.
But where Vision Pro really shines is when you combine real and virtual worlds, such as playing on a virtual chessboard placed on a table in front of you. Apple’s Encounter Dinosaur demo experience shows what’s possible by creating a portal to a prehistoric land that’s fixed to the actual wall of a room. A butterfly emerges from the portal and lands on your outstretched finger. The dinosaur then comes into view and locks eyes with you. The dinosaur’s head and eyes follow you as you move around the room, before scaring off rival dinosaurs and roaring at you.
There are only a handful of truly great experiences available on the App Store right now, but most of the best ones are controlled directly by your hands and body. Other headsets can do similar things, but none do it as easily, accurately, or with the same high fidelity as Vision Pro.
Relive your memories like never before
When you view your holiday panoramas with Vision Pro, the photos expand all around you, filling your field of vision completely for an immersive experience. Photo: Apple
One of the most unexpected and wonderful things about Vision Pro is its ability to relive past moments through photos, videos, and panoramas.
Loading a panorama photo made me feel like I was standing in Death Valley again, enjoying the dramatic colors of a sunset over the vast desert. Or sitting in a packed Capital One Arena watching the Washington Capitals play ice hockey. And a photo I took from the top of Seattle’s Space Needle gave me the same feeling of height dread I had when I took it seven years ago.
Vision Pro can also display spatial and 3D videos shot with a headset or iPhone 15 Pro. These look like the little holograms you often see in sci-fi, giving you a real sense of depth and the feeling of being back in the moment, but it takes practice to get it right
According to the CEO of the AI hardware company, the next industrial revolution has already begun. The announcement was made at a crowded stadium in Taipei on Sunday by Nvidia, as they revealed new products and plans to drive advancements in artificial intelligence.
Jensen Huang, the CEO of Nvidia, is currently attending Computex, Taiwan’s largest technology exhibition, alongside CEOs from major semiconductor companies such as AMD, Intel, and Qualcomm. The focus of the event is on establishing AI as a mainstream technology in the industry.
Having been born in Taiwan, Huang is a prominent figure on the island, and his presence has generated significant interest from the media and the public. Nvidia is recognized as the leader in specialized chips and hardware essential for the development and operation of cutting-edge AI systems.
During his address at the National Taiwan University’s Sports Center, Huang mentioned the collaboration between companies and countries with Nvidia to revamp their traditional data centers into high-speed computing facilities, focusing on creating an AI factory for the mass production of artificial intelligence.
He introduced the Nvidia ACE generation AI, which can generate lifelike human avatars for industries like customer support. Several top technology companies, including Foxconn and Siemens, are leveraging Nvidia’s platform to develop AI-driven autonomous robots.
Nvidia recently unveiled its Blackwell platform, and Huang disclosed plans to launch an “Ultra” version in 2025. He also provided a glimpse of their upcoming graphics processing unit architecture, codenamed Rubin. Huang emphasized Nvidia’s commitment to accelerating the release of new GPU products annually.
In his forward-looking speech, Huang predicted that generative AI would play a significant role in almost every interaction with the internet or computers in the future. He concluded by praising Taiwan’s advanced semiconductor industry, which plays a crucial role in manufacturing essential components for various technologies.
Keynote addresses at Computex are also expected from AMD’s Lisa Su and Qualcomm’s Cristiano Amon, outlining their companies’ plans in AI. Other speakers include Intel’s CEO Pat Gelsinger and Arm’s president Rene Haas, highlighting the significance of AI-accelerated technologies.
Taiwanese manufacturers are pivotal to technology companies’ AI strategies as they produce advanced semiconductors required for powerful AI applications. Foxconn, known for electronics production, has shifted towards AI hardware, with their CEO projecting substantial growth in the AI server market.
Despite Taiwan’s crucial role in the global supply chain, there are concerns over China’s territorial claims and potential use of force. Tensions between Beijing and Taipei have escalated, with China conducting military exercises near Taiwan, including simulated blockades.
Any university lecturer will tell you that getting students to come to their morning lectures is a real struggle.
But even the most hungover beginner is sure to be captivated by Albert Einstein’s physics lesson or Coco Chanel’s design masterclass.
This could soon be the case for students in the UK, with some universities inviting guest lecturers from around the world using the same holographic technology used to bring deceased singers back on stage. It’s starting to happen.
Loughborough University, which was the first in Europe to consider applying the technology, has used the technology to bring in sports scientists from the Massachusetts Institute of Technology (MIT) to teach fashion students how to create immersive shows and teach management students how to create immersive shows. plans to test how to handle difficult business situations.
Professor Vicky Locke, dean of Loughborough Business School and who is leading the rollout of the technology, said students “absolutely love” the technology and want to take selfies with it. said. They would prefer “a guest speaker from the industry who walks into the classroom with a smile on their face rather than a two-dimensional person on the wall,” she added.
The Zoom calls made students “feel like they were watching TV… it felt distant,” she said. For them, holographic images are more appealing and realistic. ” The technology will be officially introduced into the curriculum in 2025 after a year of experimentation.
The box-based holographic unit is sold by L.A.-based company Proto. The company’s customers include companies such as his BT and IBM, and it is used in meetings to reduce the need for business travel. We are also collaborating with Stockholm fashion retailer H&M to create an interactive product display.
David Nussbaum, who founded Proto four years ago after working on holograms of deceased celebrities, says his company could soon bring some of the 20th century’s greatest thinkers back from the dead. Told.
He added: “Proto has technology that projects images of Stephen Hawking and other people to make it seem as if he’s really there. We use it in books, lectures, social media, etc. You can connect it to something you were attached to, a question you asked, an interaction you had with him, etc. AI Stephen Hawking looks just like him, sounds like him, and interacts as if it were him. To do.
“It’s awe-inspiring and mind-blowing. I’m shocked at how great the interactions are. Whether people like it or not, AI is part of our lives.”
He added that his company’s ambition is to prove that “you don’t have to be an eccentric billionaire or celebrity to have a hologram.”
Gary Barnett, Professor of Digital Creativity at Loughborough University, who is also leading the implementation, said:
“Students need to understand what it means to use them, to be in that world, to experience them, to interact with them, and all that they will need for their future careers.”
Professor Rachel Thomson, the university’s vice-chancellor and advocate, said the technology could reduce the need to bring in guest speakers at short notice, encourage international research collaboration and reduce the amount of teaching materials used by students. He said it will help achieve sustainability strategies. Building prototypes in engineering, design and creative arts.
It also allows instructors to display complex equipment, such as engines, more easily than over a video call.
Nussbaum said corporations and large institutions such as universities are the first step in his company’s plans, but he hopes to roll out mini-units costing less than $1,000 within the next 18 months, which It will show a miniature image that he likens to the movie “Wonka Vision.” “Charlie and the Chocolate Factory” by Roald Dahl.
He added that the technology’s AI capabilities meant it was possible to create an avatar that looked like anyone in the world, but noted that this could come with legal complications.
Burning buildings, collapsing tunnels, sinking ships… wherever you're trapped and in danger, there's a robot designed to come to your rescue. Here are nine of the best.
trail blazers
Firefighting robot, Anshan Hengye Special Vehicle Manufacturing Co.
Photo credit: Getty
Firefighters are testing a fire extinguishing robot here at Bajiao Fire Station in Yantai, northeast China. The Chinese government has been promoting the technology for several years, allowing human firefighters to control robotic fire extinguishers while remaining safely outside fire danger zones.
Back in 2020, the city of Tongliao in northern China announced a firefighting mechanical unit. This 10-strong unit of his, known as the “Blade Formation”, includes robots, drones, and transport units. In other regions, ground-based robotic firefighters are being employed to fight fires in chemical plants and subways.
One of the big advantages of this approach is fire resistance. The robot can operate at temperatures of 1,000°C (1,800°F) for more than 30 minutes. Sensors can also be used to detect heat sources, difficult terrain, and flammable gases.
In the future, it may be possible to replace residential fire alarms and fire extinguishers. In 2022, Chinese researchers published the following paper: Compact and lightweight “home firefighting robot”.
go underground
Team Explorer, Carnegie Mellon University, Oregon State University
Photo courtesy of Carnegie Mellon University
Searching for people trapped in dark underground tunnels and cave complexes isn't fun for most people, but in 2017 the Defense Advanced Research Projects Agency (DARPA) turned it into a game. There may not have been any survivors, but there was a huge underground space.
Players are tasked with developing and using an autonomous robot system to search for mannequins and other objects to earn points. DARPA designed this competition to foster innovation in autonomous subsurface mapping and navigation. This could benefit rescue workers working after mine collapses or in cave rescue operations.
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