A letter from Lewis Carroll to Nature, March 31, 1887: Having hit upon the following method of mentally computing the day of the week for any given date, I send it you in the hope that it may interest some of your readers. I am not a rapid computer myself, and as I find my average time for doing any such question is…
I got an email the other day from mathematician Bogdan Grechuk, whose book Polynomial Diophantine Equations: A Systematic Approach ( was recently released. This is to my mind a rather remarkable bo…
The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Wh…
A few days ago John Graham-Cumming posted about “Steve Ballmer’s incorrect binary search interview question” which drew a lot of attention on Hacker News. The Ballmer’s favorite brain teaser goes like this:
APT 2.9.3 introduces the first iteration of the new solver codenamed solver3, and now available with the –solver 3.0 option. The new solver works fundamentally different from the old one. How does it work? Solver3 is a fully backtracking dependency solving algorithm that defers choices to as late as possible. It…
One thing you may have noticed about the trigonometric functions sine and cosine is that they seem to have no agreed upon definition. Or rather, different authors choose different definitions as the starting point, mainly based on convenience. This isn’t problematic or even particularly unusual in mathematics: as long…
What follows is an unusual story of perseverance. We start with a conjecture and after some plot twists end up discussing the meaning of truth. While the title is a spoiler, you might not be able t…
Programmers on the internet often use Turing-completeness terminology. Typically, not being Turing-complete is extolled as a virtue or even a requirement in specific domains. I claim that most such discussions are misinformed --- that not being Turing complete doesn't actually mean what folks want it to mean, and is…
Efficient filtering of polygonal shapes with closed-form solutions based on Green’s theorem with piecewise-polynomial filters (e.g. box/bilinear/bicubic).
Runescape has been a game near and dear to my heart since I was a child. Though I do not actively play anymore, it still functions as an interesting programming project substrate. Most recently, I created a bot that automatically executes trades on the Grand Exchange in order to conduct market making via common…
Here is a cheap way to obtain a set of standard weights, for measuring weights up to about 100 gram, to an accuracy of about 0.5 gram : Use the coins of the currency of your country. Thus one can make use of the fact that these coins are fabricated in mass production, and because of that have not only a constant shape…
Cosine similarity measures the angle between two vectors — the dot product divided by the product of their magnitudes. Here's what each part of the formula does.
When do music stars achieve fame, and how long does fame typically last?
Traditionally, mathematics research projects are conducted by a small number (typically one to five) of expert mathematicians, each of which are familiar enough with all aspects of the project that…
An exploration of numbers in JSON and various JSON implementations
UPDATE: I’ve written a new post with a new technique which is about 2.5x more efficient than this approach Chess notation has come a long way since descriptive notation, now we have nice and decipherable Standard Algebraic Notation, like Qxf7 (queen takes on f7) or Nf3 (knight takes on f3). This is a great text…
Summer of Math Exposition 3 - How does a computer/calculator compute logarithms?
Apple's effort to improve the UX of math.
I'm pretty sure that the 2FA codes generated by my bank's TOTP app have a bias towards the number 8 - because eight is an auspicious number. But is that just my stupid meaty brain noticing patterns where none exist? The TOTP algorithm uses HMAC, which in turn uses SHA-1. My aforementioned brain is not clever enough to…
Or how to find the largest power of 2 that divides x?
A deep dive into investigating and reducing the flutter distortion on the soundtrack of Steamboat Willie (1928).
I’ve been reading Pearl’s Causal Inference in Statistics , and one of the exercises poses this problem:
We recently made a Polylog video about a nonstandard way of understanding the P vs NP problem. As usual, our goal was to present an underrated idea in a broadly understandable way, while being slig…
You might think it’s unlikely for any interesting mathematics to arise from incense appreciation, but that’s only because you’re unfamiliar with the peculiar character of Muromachi (室町) era Japanese nobles. There has never been a group of people, in any time or place, who were so driven to display their sophistication…
An introduction to learned image compression using neural networks.
In 2019, Google and Cloudflare ran an experiment where they upgraded HTTPS for many "real users’ connections" to use post-quantum encryption. They then reported statistics on how affordable this was. Google had also run a similar experiment in 2016: "While it's still very early days for quantum computers, we're…
Backpropagation is the key algorithm that makes training deep models computationally tractable. For modern neural networks, it can make training with gradient descent as much as ten million times faster, relative to a naive implementation. That’s the difference between a model taking a week to train and taking 200,000…
In our previous post, we introduced how to utilize Discrete Event Simulation (DES) in PicoLisp. Now, let's explore another application: a railroad model…
