Mathematics students often hear about the classic “blue-eyed islanders” puzzle early in their career. If you haven’t seen it, read Terry Tao’s excellent writeup linked above. The solution uses induction and the idea of *common knowledge—*I know X, and you know that I know X, and I know that you know that I know X, and so on—to make a striking inference from a seemingly useless piece of information.
Over at Math3ma, Tai-Danae Bradley shared the following puzzle, which she also featured in a fantastic (spoiler-free) YouTube video. If you’re seeing this for the first time, watch the video first. Consider a square in the xy-plane, and let A (an “assassin”) and T (a “target”) be two arbitrary-but-fixed points within the square.
Every now and then I hear some ridiculous things about the equals symbol. Some large subset of programmers—perhaps related to functional programmers, perhaps not—seem to think that = should only and ever mean “equality in the mathematical sense.” The argument usually goes, Functional programming gives us back that inalienable right to analyze things by using mathematics.
Tai-Danae Bradley is one of the hosts of PBS Infinite Series, a delightful series of vignettes into fun parts of math. The video below is about the same of SET, a favorite among mathematicians. Specifically, Tai-Danae explains how SET cards lie in (using more technical jargon) a vector space over a finite field, and that valid sets correspond to lines. If you don’t immediately know how this would work, watch the video.
Problem: Compute distance between points with uncertain locations (given by samples, or differing observations, or clusters). For example, if I have the following three “points” in the plane, as indicated by their colors, which is closer, blue to green, or blue to red? It’s not obvious, and there are multiple factors at work: the red points have fewer samples, but we can be more certain about the position;
When NP-hardness pops up on the internet, say because some silly blogger wants to write about video games, it’s often tempting to conclude that the problem being proved NP-hard is actually very hard! “Scientists proved Super Mario is NP-hard? I always knew there was a reason I wasn’t very good at it!” Sorry, these two are unrelated. NP-hardness means hard in a narrow sense this post should hopefully make clear.
Binary search is one of the most basic algorithms I know. Given a sorted list of comparable items and a target item being sought, binary search looks at the middle of the list, and compares it to the target. If the target is larger, we repeat on the smaller half of the list, and vice versa. With each comparison the binary search algorithm cuts the search space in half.
Previously in this series: Linear programming and healthy diets — Part 1 Linear programing and the simplex algorithm Foods of the Father My dad’s an interesting guy. Every so often he picks up a health trend and/or weight loss goal that would make many people’s jaw drop.
Last week I was in Boston for the Geometry of Redistricting workshop. It was an optimistic gathering of over 500 mathematicians, computer scientists, lawyers, policy makers, teachers, and interested people of all stripes. There was a ton of information in the talks and subsequent discussions. I’ll try to distill the main ideas and avenues for research as best I can.
Problem: Express a boolean logic formula using polynomials. I.e., if an input variable $ x$ is set to $ 0$, that is interpreted as false, while $ x=1$ is interpreted as true. The output of the polynomial should be 0 or 1 according to whether the formula is true or false as a whole. Solution: You can do this using a single polynomial.
As a fun side project to distract me from my abysmal progress on my book, I decided to play around with the math genealogy graph! For those who don’t know, since 1996, mathematicians, starting with the labor of Harry Coonce et al, have been managing a database of all mathematicians. More specifically, they’ve been keeping track of who everyone’s thesis advisors and subsequent students were.