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, 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: