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Dark Alley Mathematics
Base changing and other calculations
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Tofolli gates are all you need
Any Boolean function can be computed with Tofolli gates, and Tofolli gates are reversible, and so any Boolean function can be computed reversibly.
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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 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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Arithmetic Without Numbers – How LLMs Do Math
What happens inside an LLM when it tries to calculate with nothing but matrices.
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The Intel 8087 chip: floating-point addition and subtraction
In 1980, Intel introduced the 8087, a chip that made floating-point operations much faster in the IBM PC and other systems. Although the 808...
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Rathbun's Operator
First, I’m a human typing this post. I’m not going to tell you who I am. Sure, many will say that is cowardly and fair, but I actually don’t think it would bring much value. What matters more is that I describe why, and what I did and didn’t do during the last two weeks or so with the OpenClaw agent I set up, known as…
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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…
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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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Reverse-engineering the Intel 8087's tangent algorithm: more than CORDIC
I hope you're not tired of the 8087, because I have another article about Intel's floating-point chip. 1 In 1980, Intel introduced the 8087...
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The Future of Mathematics
[This is a guest post by Jeremy Avigad. This blog post was initially written in a different file format and converted using AI. — T.] “Mathematics underwent, in the nineteenth century, …
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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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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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The Shape of Inequalities
…symmetry isn’t just a preference for “pretty” shapes. Contents Introduction The HM-GM-AM-QM Inequality The two circles The Semicircle The Container The 3D container The Sum of Squares Inequality Nesbitt’s Inequality Conclusion Introduction While I was randomly browsing the web, I came across this nice picture: Roland…
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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.
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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…
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A Trampoline
I didn’t want to buy a trampoline. It wasn’t the money.
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1 kilobyte is precisely 1000 bytes?
We usually learn that 1 kilobyte is 1024 bytes, 1 MB is 1024 kilobytes, etc. But is this true or a kilobyte is precisely 1000 bytes?
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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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Learning to solve hard problems in RL for LLMs by never giving up
This is a blog post for my recent paper on RL post-training of LLMs: introducing the Matthew Effect and proposing to solve it with Never Give Up. It is presented interactively and less formally, more like how I give the talk. For a deeper, more technical dive, check out the paper on arxiv and code on github. The…
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Heritability of human life span is ~50% when heritability is redefined
ratios will be ratios
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One night in Uzbekistan: Why was this one data point so influential?
We have a comment coming out in Nature next week that is going to cause the retraction of a high-profile paper by Kotz et al. from last year (the second most cited climate paper in the news in 2024).
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Category Theory Illustrated – Types
In this chapter we will talk about types. This might be disappointing for you, if you expected to learn about as many new categories as possible (which you don’t even suspect are really categories till the unexpected reveal)—we’ve been talking about the category of types in a given programming language ever since the…
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Six (and a half) intuitions for KL divergence
KL-divergence is a topic which crops up in a ton of different places in information theory and machine learning, so it's important to understand well. Unfortunately, it has some properties which seem confusing at a first pass (e.g. it isn't symmetric like we would expect from most distance measures, and it can be…
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Egyptian Fractions (2006)
From the highly eclectic blog of Mark Dominus
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(Post-Quantum) Cryptographic Right Answers
In the tradition started by Colin Percival and Thomas Ptacek, a set of “Cryptographic Right Answers” updated for the accelerated post-quantum transition timeline.
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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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Squares in Squares
The following pictures show $n$ unit squares packed inside the smallest known square (of side length $s$). If a pictured packing has multiple numbers in its label above, the picture represents the largest; each smaller is represented by removing any square. For the $n ≤ 324$ not pictured, the trivial packing (with no…
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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.