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November 19, 2020•4 minute read
I recently found a paper titled ‘A Cone Eversion,’ by S. Tabachnikov. I found it a bit difficult to see what was happening in 3D, so I made a visualization. Enjoy!
October 28, 2020•2 minute read
Here’s a nice solution of Putnam 1981 B5 that I haven’t seen anywhere else (so far). The main idea is to sum `bitwise,' rather than `termwise.'
Let denote the set of positive integers with the th bit set, counting from the right starting at . Then we have
A bit of thinking shows that if and only if is odd. Thus the sum becomes
September 7, 2020•2 minute read
Update 2020-09-08: This solution has been added to Kiran Kedlaya’s solution page.
I've found an elegant geometry-only proof of Putnam 2003 B5, which, to the best of my knowledge, hasn't yet been discovered.
We begin with a diagram of the problem:
August 20, 2020•3 minute read
Inspired by Matt Parker’s recent video, I decided to search for primes with , the first of which is the 46-digit
How do you go about finding more?
Well, we want to be big — very big. From high-school trigonometry, we know that this occurs when is just a tiny bit less than a half-integer multiple of . In other words, we want
August 1, 2020•9 minute read
How do you average (or sum) a lot of numbers, quickly?
More precisely, how do you do this on a webpage, preferably in real-time, when your values are stored as pixels in a large image? In general-purpose applications, we have access to the massive parallelization capabilities of GPUs with CUDA or OpenCL — but on the web, we’re stuck with WebGL and fragment shaders.
One possibility is to write a fragment shader that repeatedly downscales the image by a factor of 2, where each output pixel is the average of the four pixels in its preimage. Assuming each render takes constant time due to GPU parallelization,[1] this technique can average an texture in time.
July 29, 2020•2 minute read
It’s well-known that the period of a simple harmonic oscillator (SHO) is independent of its oscillation amplitude. But is this the only oscillator for which this holds?
No. A simple counterexample is the SHO + ‘brick wall’ potential: