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Magical Fibonacci Formulae
The following Python function computes the Fibonacci sequence , without loops, recursion or floating point arithmetic:
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The latest blog posts to take off on Hacker News. 2,851 posts.
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Scrambling eggs for Spotify with Knuth's Fibonacci hashing
In this blog post, we explore Spotify's journey from using the Fisher-Yates shuffle to a more sophisticated song shuffling algorithm that prevents clustering of tracks by the same artist. We then connect this challenge to Fibonacci hashing, and propose a novel, evenly distributed artist shuffling method.
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A new quantum algorithm for classical mechanics with an exponential speedup
Posted by Robin Kothari and Rolando Somma, Research Scientists, Google Research, Quantum AI Team Quantum computers promise to solve some problems e...
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Forecasts need to have error bars
Richard Rosenfeld in the most recent Criminologist published a piece about forecasting national level crime rates. People complain about the FBI releasing crime stats a year late, academics are wor…
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Let's try to understand AI monosemanticity
Inside every AI is a bigger AI, trying to get out
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The Dunning-Kruger effect is autocorrelation
Do unskilled people actually underestimate their incompetence?
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The largest number representable in 64 bits
A rewrite of this blog post with many new insights and updates appeared in Jan 2026
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Q* Hypothesis: Enhancing Reasoning, Rewards, and Synthetic Data
Emergency special: The information we need to understand what Q* is was right in front of us, but the memes are more fun than reality.
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How does Cave/Glade Generator Work
Watabou’s Cave Generator is one in a series of RPG-ready map generators that Watabou has created over the years. All his work oozes style, but the cave generator was always the one I found mo…
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Lindenmayer systems
A Lindemayer system or LSystem for short consists of an alphabet of symbols that can be used to make strings, a collection of production that expand each symbol into some larger string of symbols, an initial “axiom” string from which to begin construction, and a mechanism for translating the generated strings into…
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Stable Fiddusion: Frequency-domain blue noise generator
Frequency-domain blue noise generator
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B-trees in Factorio
learning about B-Trees...in factorio
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My primality testing code is faster than Sir Roger Penrose's
But, I guess he did win the Nobel Prize for Physics in 2020 so I can't really brag. However, it was a great surprise when looking at two pa...
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Is the reversal curse in LLMs real?
A recent paper The Reversal Curse points out an apparent failure in large large language models like GPT-4. From the abstract: We expose a surprising failure of generalization in auto-regressive large language models (LLMs). If a model is
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Four Kinds of Optimisation
Premature optimisation might be the root of all evil, but overdue optimisation is the root of all frustration. No matter how fast hardware becomes, we find it easy to write programs which run too slow. Often this is not immediately apparent. Users can go for years without considering a program’s performance to be an…
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37, the median value for the second prime factor of an integer
Chris Grossack's math blog and professional website.
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A linear algebra trick for computing Fibonacci numbers fast
Fast Fibonacci Computations With a Linear Algebra Twist
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Formalising modern research mathematics in real time
(This is a guest post by Bhavik Mehta) On March 16, 2023, a paper by Campos, Griffiths, Morris, and Sahasrabudhe appeared on the arXiv, announcing an exponential improvement to the upper bound on R…
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When to Shut Up: A Visual Guide (With Included Algorithm)
Use these rules to avoid boring your audience
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What do we mean by "the foundations of mathematics"?
The phrase “foundations of mathematics” is bandied about frequently these days, but it’s clear that there is widespread confusion about what it means. Some say that a proof assistant must be based on a foundation of mathematics, and therefore that the foundations of mathematics refers to some sort of formal system…
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π in Other Universes
How the value of π depends on the definition of distance.
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When gradient descent is a kernel method
Suppose that we sample a large number N N N of independent random functions f i : R → R f_i \colon \R \to \R f i : R → R from a certain distribution F \Fc F and propose to solve a regression problem by choosing a linear combination f ˉ = ∑ i λ i f i . \bar{f} = \sum_i \lambda_i f_i. f ˉ = i ∑ λ i f i . For…
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The convolution empire strikes back
ConvNets Match Vision Transformers at Scale
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Lindenmayer Systems
Originally developed by Aristid Lindenmayer in 1968, an L-System is a formalism to generate self-similar structures. Although, L-Systems are capable of generating famous fractals like the Sierpinski Triangle (Axiom: FXF--FF--FF, F → FF , X → --FXF++FXF++FXF-- , angle = 60 ) or the Gosper Curve (Axiom: FX, X →…
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Why we don’t generate elliptic curves every day
Elliptic curves are standardized, instead of being generated like Diffie-Hellman parameters. There's good reasons!
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Reconciliation is a knapsack problem
Yesterday, we announced adding AI to our recon engine. Today, we're underscoring how reconciliation is a pain point by delving into the math.