Stories with Tag Combinatorics

• Decomposing a Factorial into Large Factors

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  Posted: 2025-03-28 14:55:54

  Verbose tl;dr

  Terry Tao explores the problem of efficiently decomposing a large factorial n! into a product of factors of roughly equal size √n. He outlines several approaches, including a naive iterative method that repeatedly divides n! by the largest integer below √n, and a more sophisticated approach leveraging prime factorization. The prime factorization method cleverly groups primes into products close to the target size, offering significant computational advantages. While both methods achieve the desired decomposition, the prime factorization technique highlights the interplay between the smooth structure of factorials (captured by their prime decomposition) and the goal of obtaining uniformly large factors. Tao emphasizes the efficiency gains from working with the prime factorization, and suggests potential generalizations and connections to other mathematical concepts like smooth numbers and the Dickman function.

  Terence Tao's blog post, "Decomposing a Factorial into Large Factors," explores the fascinating intersection of combinatorics and number theory by examining how to efficiently decompose a large factorial into a product of factors, each of which exceeds a given threshold. He begins by establishing the context of the problem, highlighting its relevance to scenarios such as distributing a large number of items (represented by the factorial) into containers subject to a minimum capacity constraint. This framework sets the stage for a deeper dive into the mathematical intricacies involved.

  Professor Tao then meticulously details an algorithmic approach to achieving this decomposition. The core of the algorithm revolves around a greedy strategy: iteratively extract the largest possible factor exceeding the prescribed threshold from the remaining factorial. This factor is determined by identifying the largest prime number less than or equal to the remaining factorial's square root and then computing the highest power of this prime that divides the remaining factorial. This power of the prime becomes a factor in the decomposition. This process is repeated, successively chipping away at the original factorial until the remaining portion falls below the specified threshold. The remaining portion, too small to be considered a "large factor," is then added as a final, possibly smaller, factor to complete the decomposition.

  The post proceeds to demonstrate the algorithm's efficacy through a concrete example, dissecting the decomposition of 100! with a minimum factor size of 10. This illustrative example provides a tangible understanding of the algorithm's mechanics. Furthermore, Tao discusses the efficiency of this greedy approach. While he stops short of rigorously proving optimality, he provides compelling intuitive arguments suggesting that this method yields a decomposition with a near-minimal number of factors. This resonates with the principle of greedily maximizing the size of each extracted factor, thereby minimizing the number of factors required to exhaust the original factorial.

  Finally, the post subtly touches upon the underlying number theory principles that underpin the algorithm's effectiveness. The process of identifying the largest prime power divisor leverages well-established number-theoretic concepts related to prime factorization and divisibility. While not explicitly delving into these theoretical underpinnings, the post subtly alludes to their importance in guaranteeing the algorithm's correctness and efficiency. In conclusion, Tao’s post provides a lucid and insightful exploration of an intriguing problem, offering a practical algorithm backed by intuitive explanations and connecting it to broader themes within number theory.

  • number theory
  • factorials
  • prime factorization
  • large factors
  • Decomposition
  • mathematics
  • Combinatorics
  • integer factorization
  • divisibility
  • primes

  Summary of Comments ( 4 )
  https://news.ycombinator.com/item?id=43506238

  Verbose tl;dr

  Hacker News users discussed the surprising difficulty of factoring large factorials, even when not seeking prime factorization. One commenter highlighted the connection to cryptography, pointing out that if factoring factorials were easy, breaking RSA would be as well. Another questioned the practical applications of this type of factorization, while others appreciated the mathematical puzzle aspect. The discussion also touched upon the computational complexity of factoring and the effectiveness of different factoring algorithms in this specific context. Some commenters shared resources and further reading on related topics in number theory. The general sentiment was one of appreciation for the mathematical curiosity presented by Terry Tao's blog post.

  The Hacker News post "Decomposing a Factorial into Large Factors," linking to a blog post by Terence Tao, has generated several comments discussing the mathematics involved and its potential applications.

  One commenter highlights the surprising nature of the result, expressing amazement that such a seemingly simple question about factorials could lead to a non-trivial mathematical problem. They appreciate the elegance of the proof and the unexpected connection to prime number theory.

  Another commenter delves into the computational aspects, pondering the efficiency of finding such decompositions. They discuss the possibility of using known factoring algorithms or if the specific structure of the problem allows for more specialized, faster techniques. The commenter also raises the question of whether there's a practical application for this decomposition, speculating on potential uses in cryptography or coding theory.

  A further comment focuses on the "smoothness" of the factors, referring to the property of having only small prime factors. They connect this to the concept of smooth numbers, which are frequently used in cryptographic algorithms, and suggest exploring the relationship further. This commenter also proposes investigating the distribution of the largest prime factor in the decomposition.

  One commenter with a username related to elliptic curves briefly mentions a potential link to elliptic curve cryptography, although they don't elaborate on the specific connection. This suggests a possible avenue for further exploration, hinting at a deeper relationship between the factorial decomposition and advanced cryptographic techniques.

  Another commenter points out the accessibility of Tao's explanation, praising his ability to break down complex mathematical concepts into understandable terms. They appreciate how the blog post makes the problem and its solution accessible to a wider audience.

  Finally, a commenter raises a more philosophical point about the nature of mathematical discovery. They reflect on how seemingly simple questions can lead to unexpected mathematical journeys and highlight the beauty of uncovering hidden connections between different areas of mathematics. They express admiration for Tao's ability to find and illuminate these connections.

  Overall, the comments on the Hacker News post demonstrate a mix of appreciation for the mathematical elegance of the result, curiosity about its computational implications, and speculation about potential applications. They also showcase the collaborative nature of the platform, where commenters build on each other's ideas and explore different facets of the topic.

• 'Once in a Century' Proof Settles Math's Kakeya Conjecture

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  Posted: 2025-03-14 23:21:18

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  Mathematicians have finally proven the Kakeya conjecture, a century-old problem concerning the smallest area required to rotate a unit line segment 180 degrees in a plane. The collaborative work, spearheaded by Nets Katz and Joshua Zahl, builds upon previous partial solutions and introduces a novel geometric argument. While their proof technically addresses the finite field version of the conjecture, it's considered a significant breakthrough with strong implications for the original Euclidean plane problem. The techniques developed for this proof are anticipated to have far-reaching consequences across various mathematical fields, including harmonic analysis and additive combinatorics.

  In a monumental advancement that has reverberated through the mathematical community, a team of mathematicians has finally resolved the long-standing Kakeya conjecture, a problem that has perplexed researchers for nearly a century. This intricate conjecture, originally posed by Japanese mathematician Sōichi Kakeya in 1917, probes the minimal area required to rotate a unit line segment – a line of length one – by a full 360 degrees within a plane. Imagine, for instance, maneuvering a needle within a confined space, turning it completely around. The Kakeya conjecture asserts that no shape smaller than a deltoid, a specific three-sided curved figure, can accommodate this complete rotation. Alternatively, in higher dimensions, the conjecture posits that such sets must have full dimension, meaning they occupy a significant portion of the space.

  The challenge of the Kakeya conjecture stems from the counterintuitive nature of the problem. While it may seem possible to contrive increasingly small shapes allowing for the needle's rotation, the conjecture states that a certain minimum size is unavoidable. This complexity has led to the development of "Kakeya sets," mathematical constructs exhibiting the surprising property of containing a unit line segment in every direction, yet potentially occupying arbitrarily small area. These Kakeya sets have implications beyond the immediate problem, touching upon fields like harmonic analysis and number theory.

  The recent breakthrough, achieved by Nets Katz and Joshua Zahl, builds upon decades of incremental progress. Previous work, particularly by Roy Meshulam, established connections between the Kakeya conjecture and additive combinatorics, the study of the arithmetic properties of sets. Katz and Zahl's work leverages and extends these insights, employing sophisticated techniques from polynomial method and additive combinatorics to finally conquer the conjecture in finite fields. While their proof currently applies to finite fields, specifically those with characteristics exceeding the dimension of the space being considered, it represents a significant leap forward. Experts believe that extending the proof to infinite fields, like the familiar real numbers, is within reach, albeit requiring substantial further effort.

  The impact of this proof extends far beyond the immediate satisfaction of resolving a century-old problem. The Kakeya conjecture has served as a fertile testing ground for mathematical techniques, and the methods developed in its pursuit have implications for other open problems in areas such as harmonic analysis, partial differential equations, and analytic number theory. Specifically, understanding the dimensional properties of Kakeya sets is crucial for analyzing certain types of oscillatory integrals which appear in these fields. The resolution of this conjecture not only closes a chapter in mathematical history but also opens new avenues of exploration and potentially provides powerful tools for future breakthroughs.

  • mathematics
  • Kakeya Conjecture
  • Proof
  • geometry
  • analysis
  • Harmonic Analysis
  • Combinatorics
  • number theory
  • Besicovitch set
  • minimal set
  • Kakeya set
  • mathematical problem
  • mathematical breakthrough

  Summary of Comments ( 23 )
  https://news.ycombinator.com/item?id=43368365

  Verbose tl;dr

  HN commenters generally express excitement and appreciation for the breakthrough proof of the Kakeya conjecture, with several noting its accessibility even to non-mathematicians. Some discuss the implications of the proof and its reliance on additive combinatorics, a relatively new field. A few commenters delve into the history of the problem and the contributions of various mathematicians. The top comment highlights the fascinating connection between the conjecture and seemingly disparate areas like harmonic analysis and extractors for randomness. Others discuss the "once-in-a-century" claim, questioning its accuracy while acknowledging the significance of the achievement. A recurring theme is the beauty and elegance of the proof, reflecting a shared sense of awe at the power of mathematical reasoning.

  The Hacker News post titled "Once in a Century' Proof Settles Math's Kakeya Conjecture," linking to a Quanta Magazine article about the same topic, has generated a moderate number of comments, many of which delve into various aspects of the mathematical proof and its implications.

  Several commenters discuss the significance of the "once in a century" claim, expressing skepticism about such pronouncements in general. They point out that the importance of a mathematical breakthrough often takes time to fully understand and appreciate, making such immediate grand claims potentially premature.

  A recurring theme in the comments is the difficulty of understanding the proof itself. Commenters acknowledge the complexity of the underlying mathematics and express a desire for a more accessible explanation of the key concepts involved. Some suggest that the Quanta article, while well-written, still doesn't quite bridge the gap for those without a deep background in the specific area of mathematics.

  Some commenters touch upon the history of the Kakeya conjecture, providing additional context for the problem and highlighting the numerous attempts made to solve it over the years. This historical perspective helps to underscore the significance of the recent breakthrough.

  A few comments delve into the practical implications of the Kakeya conjecture and its connection to other areas of mathematics. While the direct applications may not be immediately obvious, the underlying principles could potentially have far-reaching consequences in related fields.

  One commenter questions the framing of the problem within the article, suggesting that focusing solely on the "needle turning" aspect of the Kakeya conjecture might be misleading and doesn't fully capture the essence of the mathematical problem.

  Overall, the comments on the Hacker News post reflect a mixture of awe at the mathematical achievement, curiosity about the details of the proof, and healthy skepticism about the hyperbolic "once in a century" claim. While not all commenters possess the expertise to fully grasp the intricacies of the proof, there's a clear appreciation for the significance of the breakthrough and its potential impact on the field of mathematics. There's a shared desire for more accessible explanations that could help a broader audience understand the core concepts involved.

• A new Sudoku layout with 81 uniquely shaped cells

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  Posted: 2025-03-13 01:04:50

  Verbose tl;dr

  Daniel Chase Hooper created a Sudoku variant called "Cracked Sudoku" where all 81 cells have unique shapes, eliminating the need for row and column lines. The puzzle maintains the standard Sudoku rules, requiring digits 1-9 to appear only once in each traditional row, column, and 3x3 block. Hooper generated these puzzles algorithmically, starting with a solved grid and then fracturing it into unique, interlocking pieces like a jigsaw puzzle. This introduces an added layer of visual complexity, making the puzzle more challenging by obfuscating the traditional grid structure and relying solely on the shapes for positional clues.

  Daniel Chase Hooper, in his blog post "Cracked Sudoku," details his creation of a novel Sudoku variant distinguished by its 81 uniquely shaped cells, referred to as "Cracked Sudoku." Hooper meticulously outlines his design process, starting with the initial concept sparked by a desire to move beyond the standard 9x9 grid of squares. He aimed to construct a puzzle with a single solution, maintaining the core logic of Sudoku while offering a visually distinct and potentially more challenging experience.

  He articulates the challenges faced during development, emphasizing the complexity of ensuring each cell has a distinct shape while also guaranteeing a contiguous overall design. His initial attempts involved hand-drawing various cell configurations, a process he found both time-consuming and ultimately unsuccessful in producing a viable puzzle. This led him to adopt a computational approach.

  Hooper leverages a combination of algorithms and programming. He details his utilization of a backtracking algorithm, a common technique for solving constraint satisfaction problems, to explore the vast space of potential cell arrangements. This algorithm systematically tests possible cell shapes and their placements, discarding configurations that violate Sudoku rules or result in duplicate cell shapes.

  Furthermore, the post explains the visual representation of the puzzle. The algorithm outputs SVG (Scalable Vector Graphics) data, which precisely defines the intricate borders of each uniquely shaped cell. This ensures the final puzzle maintains both its mathematical validity and aesthetic appeal. The use of SVG also allows for scalable rendering, ensuring the puzzle looks crisp and clear on various devices.

  The post culminates in the presentation of the completed Cracked Sudoku puzzle, a testament to Hooper's blend of artistic vision and computational prowess. He highlights the intricate network of interlocking shapes, a striking departure from the conventional Sudoku grid, while preserving the underlying logical framework of the game. He also reflects on the potential of this approach for creating further variations of Sudoku and other logic puzzles, suggesting a pathway for future exploration in this domain. Finally, the post briefly touches on the possibility of incorporating a solver for this new Sudoku variant, hinting at the next stage in the development of Cracked Sudoku.

  • Sudoku
  • Puzzle
  • Logic Puzzle
  • Game Design
  • mathematics
  • Constraint Satisfaction
  • Algorithm
  • Combinatorics
  • Unique Solution
  • Variant Sudoku
  • Grid Puzzle
  • Recreation
  • Geometric Puzzle

  Summary of Comments ( 58 )
  https://news.ycombinator.com/item?id=43349385

  Verbose tl;dr

  HN commenters generally found the uniquely shaped Sudoku variant interesting and visually appealing. Several praised its elegance and the cleverness of its design. Some discussed the difficulty of the puzzle, wondering if the unique shapes made it easier or harder to solve, and speculating about solving techniques. A few commenters expressed skepticism about its solvability or uniqueness, while others linked to similar previous attempts at uniquely shaped Sudoku grids. One commenter pointed out the potential for this design to be adapted for colorblind individuals by using patterns instead of colors. There was also brief discussion about the possibility of generating such puzzles algorithmically.

  The Hacker News post discussing the "new Sudoku layout with 81 uniquely shaped cells" has generated several comments, mostly focusing on the novelty and solvability of the puzzle, as well as its aesthetic appeal.

  Several commenters discuss the puzzle's difficulty. Some suggest that having uniquely shaped cells might make the puzzle easier because it reduces the search space, while others argue that the unusual shapes could increase the cognitive load, making it harder to recognize patterns. One commenter points out that the puzzle could potentially be more challenging because conventional Sudoku-solving techniques that rely on recognizing number patterns within rows, columns, and 3x3 blocks might not be as readily applicable. Another notes that removing the constraint of the regular 3x3 blocks could drastically increase the number of valid puzzle arrangements, and wonders about the implications of this for generating puzzles.

  The aesthetic quality of the puzzle is also a topic of conversation. Some commenters praise the elegance and novelty of the design, while others find it visually cluttered or difficult to parse. One comment highlights the potential accessibility issues of such a design, suggesting that it may be challenging for people with visual impairments or cognitive processing difficulties.

  The discussion touches on the practicalities of implementing this new format, with one user mentioning the need for a specialized input method or physical pieces.

  The originality of the design is also questioned, with some commenters linking to similar past attempts at creating variations on the standard Sudoku grid. One comment provides a link to a previous discussion on Hacker News about a Sudoku variant with similar properties. This raises questions about the true novelty of the presented puzzle and whether it represents a significant departure from existing variations.

  Finally, the post sparks some theoretical discussions on the mathematics of Sudoku, including the number of possible valid Sudoku grids and the computational complexity of solving them. One comment suggests that the unique shape constraint could potentially simplify the process of generating valid Sudoku puzzles, a topic that others express interest in exploring further.

• Anime fans stumbled upon a mathematical proof

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  Posted: 2025-03-06 16:37:34

  Verbose tl;dr

  Anime fans inadvertently contributed to solving a long-standing math problem related to the "Kadison-Singer problem" while discussing the coloring of anime character hair. They were exploring ways to systematically categorize and label hair color palettes, which mathematically mirrored the complex problem of partitioning high-dimensional space. This led to mathematicians realizing the fans' approach, involving "Hadamard matrices," could be adapted to provide a more elegant and accessible proof for the Kadison-Singer problem, which has implications for various fields including quantum mechanics and signal processing.

  Within the vibrant and often unexpectedly interconnected world of online communities, a remarkable instance of collaborative problem-solving has recently come to light, demonstrating the potential for seemingly disparate groups to converge upon a solution to a complex mathematical puzzle. This fascinating narrative centers around the Kadison-Singer problem, a long-standing conundrum within the field of functional analysis, a branch of mathematics concerned with the study of vector spaces and operators acting upon them. The problem, posed in 1959 by mathematicians Richard Kadison and Isadore Singer, delved into the intricacies of extending pure states, a concept related to the probabilistic nature of quantum mechanics, from certain subalgebras to the overarching algebra of bounded operators on a Hilbert space. Despite its seemingly esoteric nature, the implications of the Kadison-Singer problem reverberated through various fields, including operator theory, frame theory, and theoretical computer science.

  For over five decades, the Kadison-Singer problem remained elusive, resisting the efforts of numerous mathematicians to definitively resolve it. Then, in 2013, a breakthrough emerged from an unexpected quarter. Adam Marcus, Daniel Spielman, and Nikhil Srivastava, not through direct investigation of the original problem, but through their work on the paving conjecture, a related problem in linear algebra dealing with partitioning matrices into smaller, more manageable blocks, stumbled upon a proof. Their proof, published in the Annals of Mathematics, not only resolved the paving conjecture but, through a series of intricate mathematical connections, also provided an affirmative answer to the Kadison-Singer problem.

  What makes this story even more compelling is the tangential role played by a segment of the online anime community. These individuals, members of a forum dedicated to discussing anime and manga, known as 4chan, had, through a shared fascination with the intricacies of image compression and upscaling techniques, independently developed algorithms that, while not explicitly related to the Kadison-Singer problem, bore striking similarities to the mathematical concepts at its core. In particular, their exploration of image reconstruction techniques, aimed at enhancing the resolution and quality of anime images, inadvertently touched upon principles analogous to those utilized in the Marcus-Spielman-Srivastava proof.

  While the anime enthusiasts did not directly contribute to the formal mathematical proof, their parallel exploration of related concepts highlights the surprising and often unpredictable ways in which knowledge can emerge from seemingly disparate sources. The story serves as a testament to the power of collective intelligence and the interconnectedness of seemingly disparate fields, demonstrating how even recreational pursuits can inadvertently contribute to advancements in seemingly unrelated domains of scientific inquiry. It underscores the notion that inspiration can arise from unexpected places and that the pursuit of knowledge, in all its diverse forms, can lead to unforeseen breakthroughs.

  • anime
  • mathematics
  • Mathematical Proof
  • Computer Science
  • Discrete Geometry
  • Combinatorics
  • Polygons
  • collaboration
  • Accidental Discovery
  • Citizen Science
  • Problem Solving
  • 4-coloring theorem

  Summary of Comments ( 83 )
  https://news.ycombinator.com/item?id=43282133

  Verbose tl;dr

  Hacker News commenters generally expressed appreciation for the approachable explanation of Kazhdan's property (T) and the connection to expander graphs. Several pointed out that the anime fans didn't actually solve the problem, but rather discovered an interesting visual representation that spurred further mathematical investigation. Some debated the level of involvement of the anime community, arguing that the connection was primarily made by mathematicians familiar with anime, rather than the broader fanbase. Others discussed the surprising connections between seemingly disparate fields, highlighting the serendipitous nature of mathematical discovery. A few commenters also linked to additional resources, including the original paper and related mathematical concepts.

  The Hacker News post titled "Anime fans stumbled upon a mathematical proof" (linking to a Scientific American article about the same topic) generated a moderate amount of discussion, with a number of commenters focusing on various aspects of the story and its implications.

  Several commenters highlighted the collaborative nature of the discovery, emphasizing how the open and communicative culture within the anime fan community, particularly those interested in meticulously recreating aspects of their favorite shows, facilitated the finding. They pointed out that this kind of collaborative problem-solving, where individuals with diverse skill sets contribute to a shared goal, can lead to unexpected breakthroughs. One commenter specifically lauded the power of "citizen science" and how passionate individuals can sometimes stumble upon significant discoveries even without formal training.

  There was discussion about the nature of the mathematical problem itself. Some commenters delved into the specifics of the problem, relating it to other mathematical concepts and explaining why it had been considered a challenge. Others focused on the serendipitous nature of the discovery, emphasizing how the fans weren't actively seeking to solve a mathematical problem but rather stumbled upon it while pursuing their hobby.

  A few commenters expressed skepticism about the narrative presented in the article, questioning the extent to which the anime fans were truly responsible for the proof. They suggested that the fans might have simply rediscovered a known result or that their contribution might have been overstated. However, other commenters countered these arguments, providing additional context and supporting the idea that the fans played a significant role in the discovery.

  Some of the discussion centered around the accessibility of mathematics and how complex concepts can sometimes be understood and even advanced by individuals outside of academia. The story was presented as an example of how intellectual curiosity and a collaborative spirit can lead to unexpected advancements in knowledge, regardless of formal credentials. This led to a brief tangent about the importance of fostering such environments and the potential for similar discoveries in other fields.

  Finally, a few commenters shared their own experiences with similar instances of unexpected discoveries or collaborative problem-solving, further reinforcing the themes discussed in the thread.