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A decades-old bug in Knuth's long division (TAOCP Vol II, Algorithm 4.3.1D)
A long division story
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The latest blog posts to take off on Hacker News. 2,391 posts.
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Principia Mathematica is modern and insightful
Notes on Chapter 1 of Principia Mathematica
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What sort of maths are LLMs good at?
For the sake of anyone who might read this blog post in the distant future (a month from now, say), let me mention that I am writing it a few days after OpenAI announced that it had solved ten majo…
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The lattice of sets of natural numbers is rich (2021)
$\newcommand\of{\subseteq}\newcommand\N{\mathbb{N}}\newcommand\R{\mathbb{R}}\def\<#1>{\left\langle#1\right\rangle}\newcommand\Z{\mathbb{Z}}\newcommand\Q{\mathbb{Q}}\newcommand\ltomega{{{<}…
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Differential Heuristics
In 2005 Google showed off Google Maps, which lets you drag the map around instead of reloading the page like MapQuest and others used. That’s the feature that got everyone’s attention. But what got my attention was a feature they added in 2007 where you could drag the start/end points on a route [1] , and it would…
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There Are Magic Hexagons of Every Order
What is so special about the number 19?
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The Entropy of a Markov Chain
A more explicit example
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Dates That Don't Exist (2015)
In 1582, under the inter gravissimas papal bull, the Catholic world transitioned from the Julian Calendar to the Gregorian Calendar . This new calendar shortened the length of the year from 365.25 days to 365.2425 , a reduction of just 0.002% .
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A Trampoline
I didn’t want to buy a trampoline. It wasn’t the money.
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Improving Heuristics for A* Pathfinding
On my Introduction to A* page I cover the basics of the A* graph search algorithm for finding shortest paths. Here’s an example from the town of Denerim in Dragon Age Origins. Try moving the start and goal to see A* in action:
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Truth is not a direction: a Tarski attack on LLM probes
Abel Jansma is a scientist studying emergence, higher-order interactions, information theory, and complex systems across biology and artificial intelligence.
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We have proof automation now
I've long had a soft spot for dependently-typed languages like Coq Rocq and Lean. They offer the possibility of a type system capable of encoding and enforcing arbitrarily subtle invariants. The sort of thing that, in regular languages, ends up (at best) as a comment, and which quickly gets lost as the size of the…
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The Dark Night of Mathematics
What are we really doing?
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Jacobian Conjecture for Baby
An illustrated guide that starts from the basics, explains the Jacobian Conjecture, and follows what happened to it in July 2026.
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A digestion of the Jacobian conjecture counterexample
The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let $latex {F:{\bf C}^n \rightarrow {\bf C}^n}&fg=000000$ …
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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…
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What does the Riemann zeta function have to do with the distribution of primes?
One of the oldest theorems in mathematics is that there are infinitely many primes. Euclid (the geometer!) proved this around 300 BCE, and for centuries afterwards the story ended there. There are infinitely many primes -- so what else is there to ask about them?
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Backtrack-Free Cursive
Making English more enjoyable to write.
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Unicode's transliteration rules are Turing-complete
Claim: as specified, the UTS #35 transliteration-rule language is Turing-complete. This is an exception in Unicode, where algorithms are deliberately bounded and always terminate. The specification defines a revisiting cursor with no rewrite limit. ICU's implementation implements a per-call cap, but the spec itself…
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Mr. Baby Paint and accidentally discovering a new cellular automata
On building Mr. Baby Paint, a radically simple co-op drawing app for toddlers & accidentally discovering a flood-fill cellular automata, plus developing pixel-fattening algorithms for bitmap fonts.
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Parallel Parentheses Matching
The parallel parentheses matching problem can be parallelized in different manners depending on the use case and is a good example of how stacks map to parallel hardware.
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Bipartite Matching Is in NC
Since I’m a good mood today—at a beautiful science camp with my kids, high in the mountains near Big Bear Lake in California—I thought I’d blog about something positive. Las…
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Everything is logarithms
Some connections between things, which I have not seen elsewhere. Maybe they mean something?
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The case against geometric algebra (2024)
Every once in a while the internet gets talking about Geometric Algebra (henceforth GA) and how it’s a new theory of math that fixes everything that’s wrong with linear algebra and multivariable calculus. When I come across this stance I am compelled to respond with something like: “wait wait, it’s not true! GA is…
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My Mathematical Regression
i become stupid
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A Perceptron in Age of Empires II
An operational NAND gate in Age of Empires II , built with the game's scenario editor. This one can also be built within the game in a two-player setup but it's a lot easier (so to speak) to just script it.
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Data Compression Explained (2012)
Copyright (C) 2010-2012, Dell, Inc . You are permitted to copy and distribute material from this book provided (1) any material you distribute includes this license, (2) the material is not modified, and (3) you do not charge a fee or require any other considerations for copies or for any works that incorporate…
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Egyptian Fractions (2006)
From the highly eclectic blog of Mark Dominus
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Image Compression
Exploiting the quirks of human vision to make images smaller, without looking awful.
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The adder at the heart of Intel's 8087 floating-point chip
In 1980, Intel released the Intel 8087 floating-point coprocessor, a chip that could make math up to 100 times faster. As well as arithmetic...