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Human mathematicians are being outcounterexampled
It’s been an interesting few weeks for counterexamples. This post is basically my perspective of what has been going on in the world of formalization, AI tools and, in particular, counterexam…
Most cited
Posts that commenters keep linking to, ranked by how many different people have cited them in Hacker News comments. 494 posts.
302
Fermat's Last Theorem – how it’s going
So I’m two months into trying to teach a proof of Fermat’s Last Theorem to a computer. We already have one interesting story, which I felt was worth sharing.
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FunSearch: Making new discoveries in mathematical sciences using LLMs
We introduce FunSearch, a method for searching for “functions” written in computer code, and find new solutions in mathematics and computer science. FunSearch works by pairing a pre-trained LLM, whose goal is to provide creative solutions in the form of computer code, with an automated “evaluator”, which guards…
305
How to learn mathematics: the asterisk method
All original material on the geometry.org website is Copyright (C) 1999–2025, Alan U. Kennington.
306
Reverse Engineering the Hacker News Ranking Algorithm
A data-driven exploration of how the Hacker News ranking algorithm works.
307
The Waiting Time Paradox, Or, Why Is My Bus Always Late?
If you, like me, frequently commute via public transit, you may be familiar with the following situation:
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Jupyter, Mathematica, and the Future of the Research Paper
Economist. Policy Entrepreneur. Geek. Co-recipient of the 2018 Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel. University Professor at NYU. Focused on urbanization; cooperation at scale of millions and billions; science; technology; economic development; long-term growth; code as language.
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Neural network training makes beautiful fractals
This blog is intended to be a place to share ideas and results that are too weird, incomplete, or off-topic to turn into an academic paper, but that I think may be important. Let me know what you think! Contact links to the left.
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Mastering Linear Algebra in 10 Days: Astounding Experiments in Ultra-Learning
The MIT Challenge My friend Scott Young recently finished an astounding feat: he completed all 33 courses in MIT’s fabled computer science curriculum, from Linear ... Read more
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Penrose: From mathematical notation to beautiful diagrams
Materials from our SIGGRAPH '20 paper.
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Viewing Matrices and Probability as Graphs
Today I'd like to share an idea. It's a very simple idea. It's not fancy and it's certainly not new. In fact, I'm sure many of you have thought about it already. But if you haven't—and even if you have!—I hope you'll take a few minutes to enjoy it with me. Here's the idea: Every matrix corresponds to a graph. So…
315
Lost in Math
Sabine Hossenfelder’s new book Lost in Math should be starting to appear in bookstores around now. It’s very good and you should get a copy. I hope that the book will receive a lot of a…
316
Cache Eviction: When Are Randomized Algorithms Better Than LRU? (2014)
Once upon a time, my computer architecture professor mentioned that using a random eviction policy for caches really isn't so bad. That random eviction isn't bad can be surprising — if your cache fills up and you have to get rid of something, choosing the least recently used (LRU) is an obvious choice, since you're…
317
Calculate the day of the week for any date in your head
The Doomsday Algorithm gives the day of the week for any date (and you can do it in your head)
318
Number of legal 18x18 Go positions computed. One more to go
L19 = 208168199381979984699478633344862770286522453884530548425639456820927419612738015378525648451698519643907259916015628128546089888314427129715319317557736620397247064840935
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Probabilistic Filters By Example
Probablistic filters are high-speed, space-efficient data structures that support set-membership tests with a one-sided error. These filters can claim that a given entry is definitely not represented in a set of entries, or might be represented in the set. That is, negative responses are conclusive, whereas positive…
320
Is the reversal curse in LLMs real?
A recent paper The Reversal Curse points out an apparent failure in large large language models like GPT-4. From the abstract: We expose a surprising failure of generalization in auto-regressive large language models (LLMs). If a model is
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Understanding the math behind 0x5f3759df and the fast inverse square root (2012)
This post is about the magic constant 0x5f3759df and an extremely neat hack, fast inverse square root, which is where the constant comes from.
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Operation Costs in CPU Clock Cycles
Quote: Back in 80s, it was possible to calculate the speed of the program just by looking at assembly. Another Quote: keep in mind that these days compilers ...
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A Better Strategy for Hangman
Learn the optimal strategy for playing Hangman.
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LaTeX Math in MS Office
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325
How many floating-point numbers are in the interval [0,1]?
Most commodity processors support single-precision IEEE 754 floating-point numbers. Though they are ubiquitous, they are often misunderstood. One of my readers left a comment suggesting that picking an integer in [0,232) at random and dividing it by 232, was equivalent to picking a number at random in [0,1). I am not…
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Wounded QR Codes
How do QR codes work? How much is it possible to distort one and still have it readable?
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The Stack Monoid
(Updated 2020-09-06 with pointers to related work and a bit more explanation)
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Base-122 – A space efficient alternative to base-64
A space efficient UTF-8 binary-to-text encoding
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Quick, How Might the Alien Spacecraft Work?
How Stephen Wolfram helped with the theory of interstellar travel, communication with aliens and questions from the actors in “Arrival”, the movie.
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“I-Cut-You-Choose” Cake-Cutting Protocol Inspires Solution to Gerrymandering
CMU researchers say getting political parties to equitably draw congressional district boundaries can be as easy as sharing cake.