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The 8000th Busy Beaver number eludes ZF set theory (2016)
I’ve supervised a lot of great student projects in my nine years at MIT, but my inner nerdy teenager has never been as personally delighted by a project as it is right now. To…
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The latest blog posts to take off on Hacker News. 2,391 posts.
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The Fastest FizzBuzz Implementation
Benchmarks & Tips for Big Data, Hadoop, AWS, Google Cloud, PostgreSQL, Spark, Python & More...
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A Tale of Java Hash Tables
The intended audience for this article is undergrad students who already have a good grasp of Java, or seasoned Java developers who would like to explore an in-depth analysis of various hash table implementations that use Open Addressing. The reader should be familiar with Java generics, collections, basic data…
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Functorio
You might have heard people say that functional programming is more academic, and real engineering is done in imperative style. I’m going to show you that real engineering is functional, and …
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An Introduction to Lagrange Multipliers
Lagrange multipliers are used in multivariable calculus to find maxima and minima of a function subject to constraints (like "find the highest elevation along the given path" or "minimize the cost of materials for a box enclosing a given volume"). It's a useful technique, but all too often it is poorly taught and…
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The Greedy Doctor Problem
TL;DR: How to reason about people who are smarter than you. A few proposals, interspersed with reinforcement learning and humorous fiction. Ending on a surprising connection to logical inductors.
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Three sins of authors in computer science and math (1997)
Most of us learn how to write technical articles by osmosis. After reading a hundred or so papers, we unconsciously pick up the common patterns of the field and begin to imitate them. This phenomenon is largely beneficial, but there are several truly odious habits that are almost universally and inadvertently adopted…
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When did computer science theory get so hard?
I posted on When did Math get so hard? a commenter pointed out that one can also ask When did Computer Science Theory Get so Hard? For t...
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Does having prime neighbors make you more composite?
Babylonian accountants and land surveyors did their arithmetic in base 60, presumably because sexagesimal numbers help with wrangling fractions. When you organize things in groups of 60, you can divide them into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, thirtieths, and…
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What is the inverse of a vector?
If vector multiplication were invertible, we could treat vectors just like any other variable. We could do simple algebra like:
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SoME1 results
Reflections on the Summer of Math Exposition
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Seemingly impossible functional programs (2007)
Andrej has invited me to write about certain surprising functional programs. The first program, due to Ulrich Berger (1990), performs exhaustive search over the “ Cantor space ” of infinite sequences of binary digits. I have included references at the end. A weak form of exhaustive search amounts to checking whether…
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Non-Transitive Dice
We describe a set of very unusual dice, and a two player game where you will always have the advantage. You can even teach your opponent how the game works, yet still win again! Finally, we will describe a new game for three players where you can expect to beat both players at the same time! Watch James Grime and…
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Signed integers are asymmetrical
An edge case in integer safety.
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How to learn mathematics: the asterisk method
All original material on the geometry.org website is Copyright (C) 1999–2025, Alan U. Kennington.
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Sunset Geometry (2016)
Robert Vanderbei has written a beautiful series of articles and talks about a method for finding the radius of the earth based on a single photograph of a sunset over a large, calm lake.
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Solving NP-hard puzzles with the oldest trick in the book
KOITOTO hadir dengan mengutamakan kemudahan akses serta dukungan bagi pengguna yang membutuhkan informasi seputar layanan dan navigasi situs.
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Conterintuitive facts in mathematics, CS, and physics
It is possible to compute over encrypted data without access to the secret key: https://en.wikipedia.org/wiki/Homomorphic_encryption
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Maybe the spaghetti code conjecture is false
The Spaghetti Code Conjecture says that Busy Beaver programs ought to be, in some sense, as complicated as possible. Scott Aaronson puts it this way:
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Why can’t I go faster than the speed of light?
I should state from the very beginning that the title of this post is a bit of a tease. I don’t know why nothing can go faster than the speed of light. No one does. It is a fundamental pos…
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NP Complete 3rd Grade Math Problems (2018)
Leo’s third grade class got to try a Noyce Foundation math worksheet [1] the other day. They didn’t fare so well, but I gotta tell you, some of the problems are REALLY hard! In fact, on…
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The Mathematics of 2048: Optimal Play with Markov Decision Processes (2018)
Finding provably optimal strategies for 2048 using Markov Decision Processes