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Does skill affect the outcome in Snakes and Ladders?
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Have you ever played Snakes and Ladders (also known as Chutes and Ladders)? If so, are you a serious competitor?
The game traces its roots back to ancient Indian games like Pachisi, where players roll dice to progress on a square board. While Pachisi incorporates elements of luck and skill, the earliest variations of Snakes and Ladders relied solely on chance to impart a spiritual lesson about accepting one’s fate. Players advanced across a board inspired by Hindu, Jain, and Sufi teachings, cultivating virtues represented by ladders while avoiding vices symbolized by snakes.
This game made its way to the UK through families returning from British colonies. Starting in 1892, a British adaptation appeared, focusing more on simplistic morality and minimizing the spiritual aspects. Over time, moral teachings faded, leaving just the snakes and ladders.
I believe that playing a game entails making decisions that influence the outcome. In games devoid of choice, like Snakes and Ladders, the player isn’t truly engaged. If you step out of the room and someone else takes your turn, does the result change?
The randomness of gameplay can be analyzed using probability theory. A Markov chain illustrates how each step in a sequence is dictated by the probability of transitioning from the preceding position. For Snakes and Ladders, it’s possible to calculate the likelihood of landing on different spaces after rolling the dice (factoring in ladders and snakes). By analyzing all possible moves, you can determine a player’s expected position after a specified number of rolls, the estimated game duration, and other valuable statistics. Markov chains find applications across various fields in applied mathematics, including thermodynamics and population modeling.
Some games, like chess, are purely skill-based, while many others blend elements of chance and strategy. This balance significantly impacts player engagement and immersion, explaining why some favor games like Catan, which require strategic resource allocation amidst randomness, over others like Monopoly that demand fewer decisions.
For older kids who might find Snakes and Ladders monotonous, consider adding a twist: after rolling, let players decide whether to navigate up or down the board. This small adjustment enhances player interaction and engagement.
The next time you explore a new board game, ensure you’re making choices that impact the results. If not, consider pivoting to games that incorporate Markov Chains and strategic decision-making.
Peter Rowlett – A mathematics lecturer, podcaster, and author at Sheffield Hallam University, UK. Follow me on Twitter @peterrowlett
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AI Tools Revolutionize Solutions for Old Math Problems
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Amateur mathematicians are leveraging artificial intelligence chatbots to tackle historic mathematical challenges, much to the astonishment of experts. Although the questions may not represent the pinnacle of mathematical complexity, their successful resolution suggests a significant breakthrough in AI’s capabilities in mathematics, potentially altering future methodologies, according to researchers.
The challenges addressed by AI are linked to Paul Erdős, a renowned Hungarian mathematician celebrated for posing intriguing yet complex questions throughout his prolific 60-year career. “The inquiries were often straightforward but exceedingly complex,” says Thomas Bloom from the University of Manchester, UK.
At the time of Erdős’ death in 1996, over 1,000 unsolved problems existed, spanning various mathematical disciplines, from combinatorics to number theory. Today, these challenges represent critical milestones for advancements in mathematics, Bloom explains. He maintains a website dedicated to cataloging these problems and tracking mathematicians’ progress in solving them.
Given the clarity of Erdős’ problems, mathematicians began experimenting with feeding them into AI tools like ChatGPT. Last October, Bloom noted an increase in users employing AI models to uncover pertinent references in mathematical literature to aid their solutions.
Shortly thereafter, AI tools began uncovering partial improvements in results—some were previously documented while others seemed to be novel.
“I was taken aback,” Bloom recalls. “Previously, when I tested ChatGPT, it provided mere conjectures, leading me to abandon it. However, since October, I discovered genuine papers, as ChatGPT effectively analyzed existing literature, uncovering substantial insights.”
Inspired by these advancements, Kevin Barrett, an undergraduate mathematics student at Cambridge, along with amateur mathematician Liam Price, set out to identify simpler and less-explored Erdős problems amenable to AI solutions. After discovering the number 728—a conjecture in number theory—they successfully solved it using ChatGPT-5.2 Pro.
“Upon seeing the statement, I thought, ‘Perhaps ChatGPT can solve this. Let’s give it a shot,’” Barrett remarks. “Indeed, numerous experts concur that the argument is elegant and quite sophisticated.”
After ChatGPT generated the proof, Barrett and Price employed another AI tool named Aristotle, developed by Harmonic, to validate their findings. Aristotle translates traditional proofs into the Lean mathematical programming language, which is swiftly verified for accuracy by a computer. Bloom highlights this process as vital, as it conserves researchers’ limited time when confirming their results’ validity.
As of mid-January, AI tools have completely solved six Erdős problems, but professional mathematicians later identified that five of these had existing solutions in the literature. Only problem number 205 was entirely resolved by Barrett and Price without prior solutions. Additionally, AI facilitated minor improvements and partial resolutions to seven other problems that were absent in existing literature.
This predicament has sparked debate regarding whether these AI tools unveil true innovations or simply resurrect old, overlooked solutions. Bloom notes that AI models frequently need to reconceptualize problems, discovering papers that make no mention of Erdős whatsoever. “Many papers I encountered would likely have remained undiscovered without this kind of AI documentation,” he remarks.
Another point of discussion is the potential limits of this approach. While the addressed problems aren’t the most formidable in mathematics, they could typically be resolved by first-year doctoral students; nonetheless, Bloom considers the achievement significant, noting the substantial effort required for such tasks.
Barrett further emphasizes that the problems currently being solved are relatively easier compared to more challenging Erdős problems, which contemporary AI models struggle to tackle. “Ultimately, AI will need more advanced models to address complex problems,” he forecasts. Some of these challenging issues even come with cash prizes for solutions, although Barrett believes that resolutions are unlikely in the near future, stating, “I don’t think we have a model for that yet.”
Utilizing AI to tackle Erdős’ problems offers promising potential for progress, according to Kevin Buzzard. Since most of the addressed challenges are straightforward or have received scant attention, it’s difficult to gauge whether these results signify substantial breakthroughs or if they warrant professional concern. “This is progress, but mathematicians aren’t quite ready to embrace it fully,” Buzzard observes. “It’s merely a budding advancement.”
Even with the models’ current limitations, their capability to work with moderately complex mathematics could fundamentally transform how researchers craft and analyze proofs. This advancement allows mathematicians with specialized knowledge to access insights from diverse mathematical fields.
“Few individuals possess expertise across all mathematical domains, limiting their toolkit,” Bloom explains. “Being able to obtain answers rapidly, without the hassle of consulting others or investing months in potentially irrelevant knowledge, creates numerous new connections. This is a groundbreaking shift that is likely to widen the scope of ongoing research.”
It may enable mathematicians to adopt entirely novel methodologies. Terence Tao at the University of California, Los Angeles, has been instrumental in validating AI-assisted methods for solving Erdős problems.
Given their limited schedules, mathematicians often prioritize a select few difficult problems, leaving many easier yet essential questions overlooked. If AI tools can be employed instantaneously across a multitude of problems, Tao believes it could facilitate a more empirical approach to mathematics, enabling extensive testing of various solutions.
“Currently, we neglect 99% of solvable problems due to our finite resources for expert analysis,” Tao asserts. “Therefore, we often bypass hundreds of significant issues, seeking just one or two that capture our interest. We also lack the capacity for comparative studies like, ‘Which of these two methods is superior?'”
“Such large-scale mathematics has yet to be undertaken,” he concludes. “However, AI demonstrates the feasibility of this approach.”
Picture having a coffee pot that serves two cups. Poor brewing might result in a stronger brew at the bottom than at the top. When pouring from the pot into two cups, the first cup will taste much weaker than the second.
While this scenario is somewhat contrived, there are other situations where a “first is worse” (or “first is better”) approach can lead to inequity.
Consider a football game where everyone has a good idea of the skills of each player. If one team’s captain selects all players first, it creates a significant imbalance in team strength.
This scenario remains unfair even with a simple pick order. For instance, if players can be ranked from 1 to 10 based on skill, if Captain A chooses player 10 first, then Captain B selects player 9, followed by Captain A picking player 8, and so on, the resultant totals are skewed. Captain A’s team ends up with a score of 30 (10 + 8 + 6 + 4 + 2), while Captain B’s team scores only 25 (9 + 7 + 5 + 3 + 1).
So, how can we ensure a fair player selection? The answer lies in a mathematical method from the 19th century. The Tew-Morse series, initially explored by Eugène Plouet in the 1850s and subsequently detailed by Axel Tew and Marston Morse in the early 20th century, advocates for alternating and rotating choices.
In a scenario with selectors A and B, the selection order follows an ABBA pattern. The first pair is in the same order, while the second flips the order. This pattern can be extended, with a repeat that reverses the As and Bs: ABBA BAAB. Further sequences can be created like “ABBA BAAB BAAB ABBA”.
This rotation helps create equity. Using the team selection example again, the totals would be much more balanced: 10 + 7 + 5 + 4 + 1 for one team versus 9 + 8 + 6 + 3 + 2 for the other, leading to totals of 27 and 28.
An iteration of this sequence is also employed in sporting events. For instance, during a tennis tiebreak, one player serves first, followed by each player taking turns to serve two points in an ABBA sequence. This streamlined version of Tew-Morse is often seen as fairer than simple turn-taking. A similar approach is being tested by FIFA and UEFA during soccer penalty shootouts, applying pressure on the second shooter in each pair.
Returning to the coffee pot scenario, the solution is ideal. If you pour half a cup into cup A, then pour two half cups into cup B, and finally add the last half cup back into A, you will achieve equal strength in both cups. Alternatively, you could stir the coffee with a spoon. However, wouldn’t it be more gratifying to tackle such challenges with the aid of mathematics?
katie steckles – A mathematician, lecturer, YouTuber, and author based in Manchester, UK. She also contributes to New Scientist‘s puzzle column “BrainTwister”. Follow her @stex
Whether it’s military tanks, roaming wildlife, or busy cutlery in a restaurant, counting moving objects can be quite challenging. Thankfully, there exists a method that enables you to estimate the total number of items without having to count every single one.
The capture-recapture technique works by sampling. For instance, you allow some animals to roam, then collect a subset. After marking the individuals, they are returned to the population. Later, you can capture another group and count how many of them are marked.
If your first capture involves 50 marked animals, and you find that half of the second group are marked, you can deduce that approximately half of the total population is marked. Therefore, the entire population can be estimated to be around 100.
During World War II, Allied statisticians aimed to estimate the number of tanks manufactured by the German forces. Instead of releasing captured tanks, they labeled tank components with serial numbers. By recording the serial numbers of both captured and destroyed tanks, they could estimate total production under the assumption of uniform distribution. If the highest serial number recorded is l and n is the number of captured tanks, then the total tank count can be estimated as l + L/n.
For example, if the maximum serial number logged is 80, you might estimate the full range to be around 80/4 = 20, resulting in an overall estimate of about 100 tanks. This problem is commonly referred to as the German tank problem in statistics.
One of my favorite stories about estimating populations comes from a friend’s teacher. The class was tasked with estimating the number of forks in the cafeteria.
The students “captured” several forks, marking each with a spot of nail polish before releasing them back. A week later, they recaptured a sample and used it to estimate the total fork count.
Researcher executed a similar study 20 years ago. Concerned about missing teaspoons in their lab, they marked and released a number of spoons, tracked their movements, and published their findings. The outcome proved effective, prompting the notorious return of five misplaced teaspoons by the culprit in the building.
Katie Steckles is a mathematician, educator, YouTuber, and author based in Manchester, UK. She also serves as an advisor to Brent Wister, a puzzle column for New Scientist. Follow her on Twitter @stecks.
For more projects, please visit newscientist.com/maker.
Book Club shares their thoughts on Larry Niven’s Ringworld
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Experiencing the vibrant world created by Michel Nieva in his dystopian vision was something special—even more so while exploring Larry Niven’s classic science fiction work, Ringworld. Initially published in 1970, it reflects the hallmark traits of that era’s science fiction writing. While not a negative experience, it certainly offers a jarring shift for the New Scientist Book Club. Revisiting Ringworld as an adult from my teenage years prompted me to reflect on how it held up over time.
It’s important to note that much of what I cherished from Ringworld remains intact. The novel still evokes a sense of wonder, showcasing the depth of imagination, the expansive scale of its universe, and the cosmic distances it portrays. I fondly remember our protagonist, Louis Wu, at the brink of a distant planet, captivated by the Longfall River cascading into the highest waterfall in known space. His gaze followed it through the foggy mist, enthralled by the allure of the unknown.
Its grand scope is a significant aspect of why science fiction resonates with me. What insights can one gain, and what remains uncharted? The haunting imagery of razor-sharp sunflowers on Ringworld—the crew’s exploration of its vastness—created indelible impressions. Ringworld encapsulates that sense of wonder perfectly with lines like, “Men can lose their souls among the white stars… They call it A distant look. It’s perilous.”
Furthermore, I appreciated Niven’s ability to weave historical breadcrumbs into the narrative, referring to influential figures like Freeman Dyson, who inspired the concept of the Dyson sphere, as “one of the ancient natural philosophers, predating even the atomic age.” Such details are enjoyable nuggets to uncover. Additionally, Niven’s portrayal of aliens—from speakers to creatures—brilliantly evokes their essence through clever naming and design, particularly the inspiring vision of the speaker as a colossal version of our domestic cat.
As I previously mentioned, the prose does feel distinctly rooted in its time—somehow dated—with sexist undertones amidst the engaging scientific elements and intricate mathematics. The characters tend to lack depth; Louis Wu, for instance, can be quite off-putting, while Teela, our sole female character, deserves more agency. The narrative often drifts rather than following a tightly plotted journey, with characters simply moving from one event to the next without clear direction.
Intense discussions have emerged within our Facebook group, with many sharing similar sentiments. “I found enjoyment, yet felt distracted by the slow progression and the scientific facets overshadowed by the pervasive sexism,” remarked Eliza Rose, who likened it to early spy films where attractive women exist merely as accessories to the male protagonists.
Alain Pellett expressed distinct discomfort regarding Louis Wu’s treatment of women, noting that his interactions come off as unsettlingly superficial.
Gosia Furmanik, who grew up during Niven’s era, pointed out the challenge faced by non-male authors in finding supportive literary spaces. She stated, “Returning to science fiction after discovering works by authors beyond this genre’s prevalent pitfalls has been crucial,” reflecting on Ringworld in her review.
Undoubtedly, the arc of Teela’s character drew significant criticism from many readers. “I was frustrated with the conclusion of Teela’s storyline, which suggested women can only achieve significance through male figures,” wrote Samatha Lane.
Samantha also addressed a pivotal critique regarding the notion that “human males stand as the most astute beings in the universe.” This hubris roots itself in traditional humanism, positioning humanity at the center of all. This echoes the ongoing narrative surrounding our historical conquests in space—just a year after landing on the moon.
On a positive note, Niall Leighton spoke highly of the sheer scale of the novel, noting it hasn’t aged as poorly as some science fiction from that time.
Some readers appreciated Niven’s rigorous incorporation of mathematics into the narrative, stating it added an enjoyable layer to the experience. Linda Jones noted, “It has certainly enriched my enjoyment,” whereas Darren Rumbold found Klemperer Rosettes “particularly appealing.” However, not all shared this enthusiasm, as Phil Gersky commented, “I was eager to delve into this classic sci-fi novel. Unfortunately, the technobabble often marred my experience.”
Ultimately, I believe our Book Club’s exploration of this science fiction classic serves as a fascinating exercise capable of resonating with modern readers. I’m contemplating a journey into another classic soon, with suggestions pouring in from members eager to explore works by Ursula K. Le Guin, NK Jemisin, and Joan D. Vinge.
Next, we’re diving into a more contemporary read: Karian Bradley’s bestselling time travel novel, Time Saving. Yes, it features a female lead, and indeed, it passes the Bechdel Test. You can visit Karian’s site to read more about her novel and explore the intriguing opener. Join us for the discussion and share your thoughts over at our Facebook page.
They call it “the irrational validity of mathematics.” Physicist Eugene Wigner has the fascinating ability to describe and predict all kinds of natural phenomena, from the movements of planets and the strange behavior of fundamental particles to the effects of the universe, simply by manipulating numbers. He coined the term in the 1960s to summarize the facts. A collision between two black holes billions of light years away. Some are now wondering whether mathematics succeeds where all else fails, figuring out what it is that allows us to ponder the laws of nature in the first place.
That’s a big question. The question of how matter creates felt experiences is one of the most vexing problems we know of. And sure enough, the first fleshed-out mathematical model of consciousness sparked a huge debate about whether it could tell us anything meaningful. But as mathematicians strive to hone and expand the tools for looking deep within themselves, they are faced with some surprising conclusions.
In particular, they make clear that if we are to achieve an accurate account of consciousness, we must abandon our intuitions and realize that all kinds of inanimate objects, perhaps the entire universe, can be conscious. It seems to suggest that we may need to accept it. “This could be the beginning of a scientific revolution,” he says. Johannes KleinerMathematician at the Munich Center for Mathematics and Philosophy in Germany.
If so, it’s been going on for a long time. Philosophers have wondered about the nature of consciousness for thousands of years, but to little avail. Then half a century ago, biologists got involved. they discovered…
Article amended on May 4, 2020Fix: The campus of the Norwegian Inland University of Applied Sciences, where Hedda Hassel-Morch is based, has been updated to change the attribution of research on the effects of sleep or sedation on Phi.
Geometric problems involve proving facts about angles and lines in complex shapes
Google Deep Mind
Google DeepMind's AI can solve some International Mathematics Olympiad (IMO) problems in geometry almost as well as the best human contestants.
“AlphaGeometry's results are surprising and breathtaking,” says IMO Chairman Gregor Driner. “It looks like AI will be winning his IMO gold medal much sooner than was thought a few months ago.”
IMO is one of the most difficult math competitions in the world for middle school students. Answering questions correctly requires mathematical creativity, something AI systems have long struggled with. For example, GPT-4, who has shown remarkable reasoning ability in other areas, gets his 0% score on IMO geometry problems, and even a specialized AI can answer them just as well as an average contestant. I'm having a hard time.
This is partly due to the difficulty of the problem, but also due to the lack of training data. This contest has been held annually since 1959, and each round consists of only six questions. However, some of the most successful AI systems require millions or even billions of data points. In particular, geometry problems, which account for one or two out of six questions and require proving facts about angles or lines in complex shapes, are particularly difficult to convert into a computer-friendly format.
Thanh Luong Google's DeepMind and his colleagues got around this problem by creating a tool that can generate hundreds of millions of machine-readable geometric proofs. Using this data he trained an AI called AlphaGeometry and when he tested it on 30 of his IMO geometry questions, the IMO gold medalist's estimated score based on his score in the contest was 25.9, whereas the AI answered 25 of them correctly.
“our [current] AI systems still struggle with capabilities such as deep reasoning. There you have to plan many steps in advance and understand the big picture. That's why mathematics is such an important benchmark and test set in our explorations. to artificial general intelligence,” Luong said at a press conference.
AlphaGeometry is made up of two parts, which Luong likens to different thinking systems in the brain. One system is fast and intuitive, the other is slower and more analytical. The first intuitive part is a language model called GPT-f, similar to the technology behind ChatGPT. It is trained on millions of generated proofs and suggests which theorems and arguments to try next for your problem. Once the next step is proposed, a slower but more careful “symbolic reasoning” engine uses logical and mathematical rules to fully construct the argument proposed by GPT-f. The two systems then work together and switch between each other until the problem is resolved.
While this method has been very successful in solving IMO geometry problems, Luong says the answers it constructs tend to be longer and less “pretty” than human proofs. However, it can also find things that humans overlook. For example, a better and more general solution was discovered for the question from his IMO in 2004 than the one listed in the official answer.
I think it's great that you can solve IMO geometry problems in this way. Yang Hui He However, IMO problems must be solvable using theorems taught at undergraduate level and below, so this system inherently limits the mathematics that can be used. Expanding the amount of mathematical knowledge that AlphaGeometry can access could improve the system and even help make new mathematical discoveries, he says.
It's also interesting to see how AlphaGeometry deals with situations where you don't know what you need to prove, since mathematical insight often comes from exploring theorems that have no fixed proof. Yes, he says. “If I don't know what an endpoint is, can I find it in all sets?” [mathematical] Are there any new and interesting theorems? ”
Last year, algorithmic trading firm XTX Markets Total prize money: $10 million For AI math models, the first publicly shared AI model to earn an IMO gold medal will receive a $5 million grand prize, with small progress awards for major milestones.
“Solving the IMO geometry problem is one of the planned advancement awards supported by the $10 million AIMO Challenge Fund,” said Alex Gerko of XTX Markets. “Even before we announce all the details of this Progress Award, we are excited to see the progress we are making towards this goal, including making our models and data openly available and , which involves solving real geometry problems during a live IMO contest.”
DeepMind declined to say whether it plans to use AlphaGeometry in live IMO contests or extend the system to solve other IMO problems that are not based on geometry. However, DeepMind previously entered a public protein folding prediction competition to test the AlphaFold system.
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