My wife is an editor for the literary site Indie Ink. The goal of the site is to foster creativity in a non-judgemental and non-competitive manner. One way in which they do this is through a weekly writing challenge, where authors write in response to prompts given by other authors. Every once in a while, when I have some free time, I sign up to play. This happens to be one of those weeks.
In the past, we have spent a significant amount of time discussing the Mandelbrot set and variations thereon. We saw that each Mandelbrot, multibrot, and Julia set can be characterized as a set of complex numbers that behave “nicely” with respect to a function of the form \(f(z) = z^e + K\), where the choice of \(K\) gives us either a multibrot or Julia fractal. These functions are only a very small subset of all possible complex valued functions, and there are many interesting fractals that can be rendered by examining the behaviour of other kinds of functions. In this post, we will examine a function with a relationship to an open problem in number theory called the Collatz conjecture.
I don’t know a lot about the game Sudoku—I haven’t played that may squares, nor have I ever gotten really into the mathematical study of the game. However, I can certainly understand the attraction: there are a large number of interesting questions that can be asked about Sudoku that are easily stated, but require some pretty nifty mathematics to come to terms with.
