Adding probabilities and velocities have at least two things in common — a maximum and Pascal’s triangle.

Pascal’s Triangle Animation from Wikipedia

Introduction

Firstly, I know I will be disappointing some but reassuring others that this is not about Relativistic Quantum Mechanics. If you are comfortable with basic algebra, you should find it fairly easy to follow this article. If you’re already familiar with Pascal’s triangle, please skip the next section.

What is Pascal’s Triangle ?

Pascal’s triangle is an pictorial algorithm for obtaining the coefficients for binomial expansions of various powers. …


I watched a Numberphile video a while back that posed a problem about decelerating cars that has a surprising and scary answer. Two cars are exactly side by side on a multi-lane road when their drivers break to avoid an obstacle coming up and both cars decelerate at exactly the same constant rate. One car is traveling at 70 mph and manages to stop just before the obstacle. The other car is traveling at 100 mph. What speed does the second car hit the obstacle at?

Diagram 1: Two cars emergency breaking to avoid a tree on the road

The answer seems as if it should be 30 mph. While it is true…


Diagram 1: A Chess Board where a key is placed in a secret compartment in one square. For the purposes of this article, we’ll be assuming the 3rd column of the 2nd row.

Introduction

This article flows out of a video conversation between the YouTubers and mathematicians Grant Sanderson and Matt Parker (references below) and an email conversation between myself and a friend of mine: Peter Gaffney as we explored our own solutions to this beguiling puzzle.

Exploring different approaches to this seemingly impossible problem reveals some interesting connections between various branches of mathematics including logic, set theory, multidimensional vector spaces and even category theory.

Outline of the Puzzle

You and a fellow prisoner are imprisoned in a dungeon and are facing execution. As is usual in these problems, the prison warden has both a love of recreational…


Screenshot from the Polyhedral Animator

Ancient history but I wrote an article way back when for Microsoft’s MSDN magazine. Because Silverlight was strictly 2D at the time, there was an opportunity to leverage a lot of the mathematics that nowadays is hidden away in the GPU hardware or the 3D canvas in HTML5.

Silverlight is no longer with us but mathematics is of course eternal and I include a link in case it’s of interest for other animation or education projects:

https://docs.microsoft.com/en-us/archive/msdn-magazine/2008/april/silverlight-building-advanced-3d-animations-with-silverlight-2-0


For those who want more than speculation, here's the Wikipedia article (complete with references):

https://en.wikipedia.org/wiki/Salvatore_Pais

Spoiler: It's not looking good for the "Pais" effect,


Identity and Freedom

Mandelbrot Set from WikiMedia Commons

Most sensible people accept that there is an objective reality out there, albeit wonderfully complex. Once we start assigning labels, things become more subjective because we have to choose what little to take into account and ignore everything else. Moreover the categories we choose to make sense of reality may seem simple from a distance but almost invariably end up with endlessly elaborate fractal “coastlines” the closer we look.

Categories such as “living” or “species” have such boundaries that are open to interpretation where different experts disagree and Pluto can sail on through the firmament completely unaffected by a vote…

Declan Brennan

Software Engineer, Entrepreneur, Lover of Science and Diversity in all its forms, Skeptic.

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