Infinite Fractal Mazes

My previous post, “Solving fractal mazes” is a prerequisite to this one. Fair warning, this will be long and dense.

Fractal mazes contain infinite paths, but the only solutions permitted are finite. Some people find that disappointing. What’s the point of all that extra maze if we don’t get to traverse it? So my goal is to come up with a variant ruleset for fractal mazes that permits and formalizes infinite solutions. In fact, I will propose two distinct rulesets, provocatively titled Countably Infinite Fractal Mazes and Cantor Fractal Mazes.

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Solving Fractal Mazes

What are fractal mazes?

Fractal Mazes are a type of maze popularized (or invented?) by Mark J. P. Wolf, published in the Mathpuzzle blog in 2003. A fractal maze is a maze that contains nested copies of itself.

small fractal maze

“Small Fractal Maze”. Credit: Mark J. P. Wolf. Source: Mathpuzzle

Fractal Mazes are typically visually represented as a sort of circuit diagram. In the above image, the goal is to find a path between the “+” and “-” by following the colored wires. The wires are color coded in order to clearly indicate where paths cross over/under each other. The three modules, labeled A, B, and C, are each copies of the entire maze. However, the start and finish only exist in the largest copy of the maze. So however deep you go into the fractal, you must eventually climb all the way back up again.

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Link Roundup: October 2023

A Weird New Scam | stderr – Remember when FTB went down for a few days around September 12 or so?  Marcus Ranum explains what happened, entertainingly.  The short version is, someone claimed they had a copyright on the banner image–you know, the one that says freethoughtblogs.com on it–and the hosting service shut down the site because DMCA is fundamentally broken.

Fractal Mazes – Commenter amito pointed me to their fractal maze browser app (see app, Github).  (Solver beware: I’m pretty sure the first maze by Noke Lieu just doesn’t have a solution.)  And then Jay McArthur linked to their own github page with a collection of more fractal mazes with citations, plus a python app.  I’m proud to have made three mazes featured in both of these.  I made them a long time ago (here’s one), but they’re still kicking around.

Fractal mazes are great, I love them.  I first heard about fractal mazes in 2003 through MathPuzzle, and then I designed three myself almost a decade ago.  You cannot solve fractal mazes with the conventional right-hand rule, but there is a computationally efficient terminating algorithm that will solve any fractal maze.  Perhaps one day I will describe the algorithm.

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The Random Number Game

I invented a game, and it goes like this. We’re going to pick a 20 digit number by taking turns choosing each digit. I choose the first digit, then you choose the second digit, I choose the third and so on. Once we’ve chosen all the digits, we use our number as the seed to a random number generator. The random number generator picks a number between 0 and 1, and if the number is greater than 0.5 then I win; if it’s less than 0.5 then you win.

Obviously this isn’t meant to be a “fun” game, it’s more of an open-ended math problem. What’s the strategy? Is there a strategy? Who wins?

The idea behind the random number generator, is that it’s deterministic, and yet opaque. Given any particular seed, the random number generator will consistently pick the same result—either you win, or I do. But there’s no particular pattern to it. It behaves as if the result were randomly chosen. The only way to predict the game’s outcome is to individually plug in each random seed into the random number generator. However, this might be intractable, as there are 10^20 possible seeds.

This game is deterministic, finite, and perfect information—much like Chess. However, it appears that the only real strategy is brute force, by plugging in seeds into the random number generator.

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Origami: Water drops

Water Drop, by Ekaterina Lukasheva

For quite some time, I had been complaining about lack of origami books about curved crease techniques.  But now we have one!  Curved Origami by Ekaterina Lukasheva has a bunch of curved crease projects from very simple to moderately complex.  This is one of the first models in the book.

For those who don’t know, a “curved crease” is a crease that makes a curved line on the paper.  A curved crease will not fold all the way, meaning that the folded models necessarily make a 3-dimensional shape.  The typical folding method involves drawing and scoring curved lines on paper (possibly with the assistance of templates, compasses, french curves).

I really admire how simple these water drops are.  (Yeah, I know rain drops aren’t actually shaped like that.)