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The largest number representable in 64 bits
This post is a rewrite of my earlier blog post from Nov 2023 with new insights and updates.
New
The latest blog posts to take off on Hacker News. 2,391 posts.
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Dark Alley Mathematics
Base changing and other calculations
153
Nonograms: a practical guide with interactive examples
This is a very important technique: only fill cells when you’re certain; no guessing.
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Case study: Creative math – How AI fakes proofs
Many AI enthusiasts debate whether Large Language Models actually "reason." My research indicates that a reasoning process does indeed occur, but its goal is different than we assume.
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Golden Ratio using an equilateral triangle inscribed in a circle
Another interesting way to graphically derive the Golden Ratio using an equilateral triangle inscribed in a circle and the chord which passes through the midpoints of the sides of the equilateral triangle.
156
Nvidia Stock Crash Prediction
One of the questions of the 2026 acx prediction contest is whether Nvidia’s stock price will close below $100 on any day in 2026. At the time of writing, it trades at $184 and a bit, so going down to $100 would be a near halving of the stock value of the highest valued company in the world.
157
Floating-Point Printing and Parsing Can Be Simple and Fast
A floating point number f has the form f = m · 2 e where m is called the mantissa and e is a signed integer exponent . We like to read numbers scaled by powers of ten, not two, so computers need algorithms to convert binary floating-point to and from decimal text. My 2011 post “ Floating Point to Decimal Conversion is…
158
The microstructure of wealth transfer in prediction markets
Slot machines on the Las Vegas Strip return about 93 cents on the dollar. This is widely considered some of the worst odds in gambling. Yet on Kalshi, a CFTC-regulated prediction market, traders have wagered vast sums on longshot contracts with historical returns as low as 43 cents on the dollar. Thousands of…
159
Starting from scratch: Training a 30M Topological Transformer
Tauformer is a topological transformer (see paper ) that replaces dot‑product attention with a Laplacian-derived scalar (taumode) per token/head, then attends using distances in that scalar space. Below is a post-style overview of the idea and the first training signals from a 30M-parameter run.
160
All 23-Bit Still Lifes Are Glider Constructible
In the Game of Life, which still lifes can be produced by crashing gliders together? We’ve known for a few years that the answer cannot be “all of them”, because in 2022 Ilkka Törmä and Ville Salo found a patch of still life that, if it exists in the universe, must have existed since the beginning of time. And so…
161
How to store a chess position in 26 bytes (2022)
Made in Notion, the connected workspace where AI agents and teams organize docs, projects and knowledge together.
162
The unreasonable effectiveness of the Fourier transform
New : You can now watch a recording of my talk on my YouTube channel ! Or you can just click "play" below, I suppose.
163
The Q, K, V Matrices
At the core of the attention mechanism in LLMs are three matrices: Query, Key, and Value. These matrices are how transformers actually pay attention to different parts of the input. In this write-up, we will go through the construction of these matrices from the ground up.
164
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…
165
Attention Is Bayesian Inference
My journey from building cricket bots to building “wind tunnels” with my collaborators — and finding the answer hidden in geometry.
166
Parsing Advances
I find myself writing yet another toy parser, as one does during a Christmas break. It roughly follows Resilient LL Parsing Tutorial. Not because I need resilience, but mostly because I find producing a syntax tree and a collection of diagnostics a more natural fit for the problem than bailing out on the first error.
167
How Lewis Carroll computed determinants (2023)
Charles Dodgson (a.k.a. Lewis Carroll) developed a method of computing determinants which has some practical advantages.
168
2D Signed Distance Functions
Tutorials and articles of Inigo Quilez on computer graphics, fractals, math, art, demoscene, shaders and more.
169
More on whether useful quantum computing is “imminent”
These days, the most common question I get goes something like this: A decade ago, you told people that scalable quantum computing wasn’t imminent. Now, though, you claim it plausibly is immi…
170
What is an elliptic curve? (2019)
The informal definition of an elliptic curve is simple. The formal definition is more involved.
171
Erdős Problem #1026
Problem 1026 on the Erdős problem web site recently got solved through an interesting combination of existing literature, online collaboration, and AI tools. The purpose of this blog post is to try…
172
From Zero to QED: An informal introduction to formality with Lean 4
An informal introduction to formality in Lean 4
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50 years of proof assistants
Crackpots ranging from billionaire Peter Thiel to random YouTube influencers claim that science has been stagnating for the past 50 years. They admit that computing is an exception: they don’t pretend that my personal 32GB laptop is not an advance over the 16MB mainframe that served the whole Caltech community when I…
174
When would you ever want bubblesort? (2023)
There are very few universal rules in software engineering, but there are are a lot of near-universal principles. Things like "prefer composition to...
175
Beautiful Abelian Sandpiles
I can't remember where I first saw them, but ever since, I have been unable to forget them: abelian sandpiles. I'm far from the only one . They're remarkably simple, yet produce lovely symmetric patterns. I loved them so much that I adorned the title banner of this blog with an animation of an abelian sandpile. But…
176
My favourite small hash table
I'm the kind of person who thinks about the design and implementation of hash tables. One design which I find particularly cute, and I think deserves a bit more publicity, is Robin Hood open-addressing with linear probing and power-of-two table size. If you're not familiar with hash table terminology, that might look…
177
Learning Feynman's Trick for Integrals
Among a few other integral tricks and techniques, Feynman's trick was a strong reason that made me love evaluating integrals, and although the technique itself goes back to Leibniz being commonly known as the Leibniz integral rule, it was Richard Feynman who popularized it, which is why it is also referred to as…
178
Zero knowlege proof of compositeness
Testing whether a number is composite as an example of a zero knowledge proof. How ZKPs are used more generally.
179
A triangle whose interior angles sum to zero
In hyperbolic geometry, the sum of a the interior angles of a triangle can be any value less than π. If the sum is zero the triangle has area π.
180
The inefficiency of RL, and implications for RLVR progress
And implications for RLVR progress