The knot problem for mathematicians finally has a solution
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Why is it trickier to untie two small knots compared to one large knot? Surprisingly, researchers have found that larger, seemingly complex knots formed by combining simpler ones are, in fact, easier to untangle. This discovery contradicts notions held for nearly 90 years.
“We were searching for counterexamples without anticipating we’d actually find one, as this speculation has persisted for so long,” Mark Brittenham from the University of Nebraska, Lincoln, shared. “In the back of our minds, we thought the speculation was likely right. It was an unforeseen and astonishing outcome.”
Mathematicians like Brittenham study knots by considering them as intertwined loops with connected ends. A fundamental principle in knot theory is that each knot has a “knot number,” representing the instances a string is cut, with another segment inserted and rejoined at a junction known as a “note.”
Calculating knot numbers can be computationally demanding, with certain knots containing 10 intersections remaining unsolved. Thus, analyzing knots by breaking them down into two or more simpler knots is often advantageous. This concept is akin to prime numbers in number theory.
Now, alongside Susan Hermiller at the University of Nebraska, Lincoln, Brittenham demonstrates that this may not be the case. “This speculation has lingered for 88 years; as people failed to disprove it, the desire for it to be true persisted,” Hermiller noted. “Initially, we uncovered one example, soon revealing an infinite number of knot pairs where the number of knots was strictly less than the total for the two knots combined.”
“We discovered that our understanding was not as clear as previously thought,” Brittenham remarked. “Even knots that lack connections may untie more efficiently than we expected.”
Examples of knots that are easier to undo than components
Mark Brittenham, Susan Hermiller
Finding and verifying counterexamples involves a mix of existing knowledge, intuition, and computational strength. Remarkably, the final proof verification was achieved through a straightforward, practical approach: tying knots with a rope and physically demonstrating their resolvability.
Andras Juhasz from Oxford University, who previously collaborated with AI firm DeepMind to validate various knot theory speculations, attempted to solve this latest challenge similarly but faced no success.
“We spent a year or two seeking counterexamples without luck, so we eventually abandoned the effort,” Juhasz mentioned. “AI might not be the best tool for finding counterexamples, akin to searching for needles in haystacks – a profoundly elusive pursuit.”
Applications of knot theory vary widely, spanning from encryption to molecular biology. Nicholas Jackson at the University of Warwick in the UK cautiously suggests that this new development could have practical implications. “We seem to have gained a deeper understanding of how circular entities operate in three-dimensional spaces than we did previously,” he remarked. “Concepts that were unclear a few months ago are now coming into clearer view.”
and othersLike many of you, I deeply resent the insidious penetration of subscription services. I started with an affordable, shareable Netflix subscription years ago. Then Spotify, then Disney+ when I had kids, then Prime Video, all of which somehow made sense. Then Fitbit started charging me to unlock features on a device I’d already bought. Google now charges me a monthly fee to store the photos I take with my Google phone in its cloud. I pay a yearly fee for an app that lets me look at guitar tablature. Last week, I tried to buy protein powder only to find I could only buy it if I committed to at least a three-month supply. That’s awful.
When it comes to gaming, I’ve been an Xbox Live subscriber on and off since 2003. PlayStation Plus came later, and Nintendo Online came much later with the launch of the Switch. I don’t play many live service games, but otherwise I would have paid £8.99 for the Battle Pass. To add to this already difficult situation, last week Microsoft announced an update to their video game subscription offers that required a spreadsheet to understand.
Currently, there are two words, Game Pass Standard and Game Pass Core, but they both mean the same thing. There is also Game Pass Ultimate, and some of these options apply to PC as well, while others don’t. Some offer new Microsoft games from day one, some don’t. Some include cloud gaming, some don’t. And they all have higher prices now. I’m not stupid, so even after going through these options multiple times, I can’t 100% explain what they mean and how they differ without referring to a table.
“It’s like an Animal Crossing tax.” Photo: Nintendo
PlayStation Plus is now almost as confusing and expensive. The options are called Plus Premium, Plus Extra (also synonyms), and Plus Essential, and the linguistic clarity is enough to make you want to scream into a pillow. They all come with different perks, but you can’t play online with friends without paying. As for Nintendo Switch Online, well, it has almost no perks, so it feels like a tax on Splatoon 3 or Animal Crossing. But at least it’s simple, with only two options, and significantly cheaper than Xbox or PlayStation.
I hate feeling like I’m paying a small mortgage to every entertainment and services company in existence. You could argue that every company is an option, but it doesn’t feel that way. And outside of gaming, most of these services have proven to capture the market with a relatively cheap introductory price, get everyone to switch, and then raise prices once their competitors fall far enough behind. This is why I’m skeptical of things like Game Pass in general. At the moment you can’t claim it’s a very generous offer with a great games library, but if in 10 years’ time Microsoft has bought up the games industry further and decided to charge you £30 a month to play Call of Duty, you’re going to be in trouble.
I often affectionately tease my partner about his unwavering attachment to physical media. In addition to a record collection that’s thousands of records long and shelves of Blu-rays and DVDs, he also buys boxed games on discs and cartridges like it’s 2005. But his approach is starting to seem like an act of rebellion: At least, despite the decline of video game retail, there’s still the option to actually buy and own games.
He’ll be the only one laughing when, in future, I pay £100 a month for an Ultra Super Game Cloud Box Plus to access and play my digital library of games I bought 10 or 20 years ago.
What to Play
Short, simple and anxiety relieving…Flock. Photo: Annapurna Interactive
Under review Swarm This week’s game is super simple, short and anxiety-relieving. Ride your bird through colorful and bizarre landscapes, identifying and recruiting wildlife. They all look a bit like flying fish, but you’ll know the difference between a piper and a greaves, a bell and a droop, and some animals are hiding. Really It blends in well with the environment and feels like a puzzle game, where you’re figuring out how to find a creature based on a sentence from a field guide.
I wish there was more to the game – the creatures that follow you around don’t do much, for example – but it’s relaxing and stylistically interesting.
Available on: PlayStation 4/5, Xbox, PC Estimated play time: 5 hours
What to Read
Prince Faisal bin Bandar bin Sultan Al Saud, chairman of the Saudi Esports Federation. Photo: Anadolu/Getty Images
I reported last year that the Olympics had tried but failed to include esports. The IOC has now He signed a 12-year contract New event series with Saudi Arabia eSports OlympicsSaudi Arabia is already hosting an esports World Cup, and its Savvy Games Group has made huge investments in various gaming companies, all of which can be seen as an extension of the country’s broader sports-washing efforts.
Listen up, all you old-school Roller Coaster Tycoon fans. Frontier has announced Planet Coaster 2A sequel to a highly detailed theme park simulation You will also be able to build a water park..
MobileGamer.biz speculates that: Only 2,000 people have paid Resident Evil 7The reported revenue figures suggest this is a technically impressive iOS port, which, if correct, raises the question of whether the market for premium console-style games on smartphones even exists.
Nintendo in Tokyo, Japan. Photo: Asker Karimalin/Alamy
The well of questions is drying up, so please send them in. I dug through the email archives for this one. Luke:
“With all of this, with game industry layoffs, the state of console gaming, and late capitalism, where does Nintendo fit in? All this? Big Because 3 is primarily a games company and not a division of a large tech company, they have quietly continued to operate without any major hiring or firings. Is this a by-product of their business culture, or something else?
There are many reasons why Nintendo is particularly resilient: it has large cash reserves, sells consoles at a profit rather than at a loss on hardware to make money on games, and has very high staff retention, allowing for the transfer of organizational knowledge. The company’s much-missed former president, Satoru Iwata, made headlines during the Wii U era when he cut his own salary to protect staff from layoffs. But this is not unusual: Japanese companies generally do not hire and fire employees repeatedly due to employment laws.
This article Gaming Industry Let me explain the labor protections that Japanese developers enjoy. It’s nearly impossible to fire employees unless a company is on the brink of bankruptcy. And this is just one of the many reasons why Japanese companies are not affected by the current flurry of layoffs. Thanks to investment from China and the size and profitability of the mobile games market, the games industry is stable and actually growing in Japan.
If you have any questions for the Question Block or any other comments about the newsletter, Please email us at pushingbuttons@theguardian.com.
Bumblebees may be capable of advanced social learning
David Woodfall/naturepl.com
Bumblebees can teach each other how to solve puzzles that are too difficult for them to solve alone. This finding suggests that these insects may use advanced social learning that has previously been demonstrated only in humans.
Previous research by alice bridges Queen Mary University of London has proposed that bumblebees could teach each other how to open lever puzzles to obtain sweet treats. And they preferred solutions they learned from their peers to solutions they had come up with on their own, as if the techniques were a cultural trend.
Now, Bridges challenged the bees to a more difficult puzzle box that required them to operate a blue lever and then a red lever in order. Even after 12 to 14 days of trying, the bees from three different colonies couldn’t figure it out on their own.
The researchers then taught nine bumblebees the key. But the training was so difficult that the animals initially refused to participate until the humans provided additional sweet rewards along the way, Bridges said. Once reintroduced to the colony, the skilled bee passed on its new knowledge to five other bees who had never seen the puzzle box before.
“suddenly, [naive bees] We were able to learn everything from trained demonstrators,” Bridges said. “When I could barely train, [the demonstrators] To do that. “
Until now, there was little evidence that non-human animals are capable of cumulative culture (defined as the ability to learn skills from other animals that cannot be acquired through a lifetime of independent trial and error). This feat allowed humans to create complex knowledge systems like modern medicine.
These findings “raise serious questions about this idea of human exceptionalism,” they wrote. alex thornton At the University of Exeter, UK his explanation on paper.
But we shouldn’t praise the cumulative culture of bees just yet. Elisa Bandini At the University of Zurich. She is not convinced that the experiment shows a behavior so complex that individual bees cannot develop it on their own. If the untaught bees had received additional rewards in the same way as the trained bees, they might have solved the puzzle on their own.
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