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100 Years to Solve an Integral (2020)
The integral of sec(x) is well known to any beginners calculus student. Yet this integral was once a major outstanding maths problem. It was first introduced...
Topics /
Math & Algorithms
The most popular blog posts about math & algorithms on Hacker News. 212 posts.
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500 days of math
I recently crossed 500 days of practicing math daily with Math Academy. I wrote about my experience after 100 days here. TL;DR: I am still very impressed by the Math Academy system and highly…
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Solving Fizz Buzz with Cosines
Fizz Buzz is a counting game that has become oddly popular in the world of computer programming as a simple test of basic programming skills. The rules of the game are straightforward. Players say the numbers aloud in order beginning with one. Whenever a number is divisible by 3, they say 'Fizz' instead. If it is…
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Is chain-of-thought AI reasoning a mirage?
Reading research papers and articles about chain-of-thought reasoning makes me frustrated. There are many interesting questions to ask about chain-of-thought…
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How Lewis Carroll computed determinants (2023)
Charles Dodgson (a.k.a. Lewis Carroll) developed a method of computing determinants which has some practical advantages.
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Think in math, write in code (2019)
Programmers love to discuss programming languages. We not only debate their technical merits and aesthetic qualities, but they become integrated into our personal identities, along with the values and traits that we associate with them. Some even defend a form of Linguistic Determinism that thinking is confined to…
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Discovering errors in Donald Knuth's TAOCP
As Donald Knuth has just published Volume 4, Fascicle 7 of The Art of Computer Programming (TAOCP), on constraint satisfaction, I would like to provide…
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Minesweeper thermodynamics
You know how sometimes you start a game of Minesweeper and immediately get stuck?
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Generating Voronoi diagrams using Fortune's algorithm
This article is about using Fortune’s Algorithm to generate Voronoi Diagrams in O(nlogn) time. If I’d known how hard it would be I wouldn’t have started it. If you are intending to implement it with the intent to actually use it, rather than as an exercise, I would recommend not using Fortune’s. Unless you are doing a…
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How Slide Rules Work
The survival of our species owes much to our brain, specifically, its ability to observe, analyse, and plan. Planting crops and storing grains for the winter were some of the earliest uses of these abilities. Measuring and calculating are foundational elements of observation, analysis, and planning. Computation, upon…
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Gaussian integration is cool
Brief discussion on gaussian quadrature and chebyshev-gauss quadrature
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Very Wrong Math
The difference between misinformation and disinformation is the difference between ignorance and malice. Trolling is somewhat different, incorporating an element of provocation and narcissism. But what the hell is this?
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Reinforcement learning, explained with a minimum of math and jargon
To create reliable agents, AI companies had to go beyond predicting the next token.
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Who Can Understand the Proof? A Window on Formalized Mathematics
Stephen Wolfram discusses understanding proofs discovered using automated theorem proving. Based on Wolfram’s proof of the simplest axioms of logic of Boolean algebra.
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Relaxed Radix Balanced Trees (2024)
I’m adding immutable vectors to my language, Ivan, and needed to pick a suitable data structure to implement them. Clojure uses Persistent Vectors (PVs) which support lookups, updates, and appends all in ‘effectively’ constant time. However, it doesn’t have efficient insert or merge operations. Relaxed Radix Balanced…
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Memory access is O(N^[1/3])
In computer science, we often compare the efficiency of algorithms by describing their runtime as a function of the size of the input. Sorting is O(n * log(n)), meaning that sorting a list of N items takes an amount of time proportional to the number of items multiplied by its logarithm. Matrix multiplication is…
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Are polynomial features the root of all evil? (2024)
High-degree polynomial features aren’t inherently unstable: the power basis and unscaled inputs are. This post shows how regularization and Bernstein polynomials make polynomial regression behave.
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A new Sudoku layout with 81 uniquely shaped cells
My journey making a Sudoku variant that doesn’t have rows or columns
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A visual exploration of vector embeddings
For Pycon 2025, I created a poster exploring vector embedding models, which you can download at full-size . In this post, I'll translate ...
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Mandelbrot deep zoom theory and practice (2021)
Perturbation and rescaling for efficient rendering of fractal images.
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Why Does Integer Addition Approximate Float Multiplication?
Here is a rough approximation of float multiplication (source): float rough_float_multiply(float a, float b) { constexpr uint32_t bias = 0x3f76d000; return bit_cast<float>(bit_cast<uint32_…
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Spline Distance Fields
Earlier this year I decided to put my main project Tangerine on hold indefinitely and began prototyping a new renderer in an unfamiliar language using a totally new graphics API with the goal of overcoming all of Tangerine's technical shortcomings. The new renderer is an eccentric CPU ray tracer called Star Machine…
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The Unknotting Number Is Not Additive
On June 30, 2025, Mark Brittenham and Susan Hermiller uploaded a preprint to the arXiv called “Unknotting number is not additive under connected sum” (and an updated version on Septembe…
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210 IQ Is Not Enough
You will never have enough IQ, nor money, nor time, nor energy.
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The most famous transcendental numbers
The mathematical constant pi represents the ratio of the circumference of a circle to its diameter. It is the most famous ratio in mathematics both on Earth and probably for any advanced civilization in the universe. The number pi, like other fundamental constants of mathematics such as e = 2.718..., is a…
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From multi-head to latent attention: The evolution of attention mechanisms
From Multi-Head to Latent Attention: The Evolution of Attention Mechanisms What is attention? In any autoregressive model, the prediction of the future tokens is based on some preceding context …
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Finding Atari Games in Randomly Generated Data
Discovering Atari Games in Randomly Generated Data
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How much oranger do red orange bags make oranges look?
Look at this orange: Now look at this orange: It’s the same orange. But, look how much more orange it looks with the red mesh on top of it: If you buy bags of oranges (at least at many places in the US), they frequently come in this red mesh bag. This bag makes the oranges look more orange. Oranger. Here’s what that…