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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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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How to hear FM Synthesis

A big reason why I enjoy dabbling in electronic music production is that it’s a way of directly transforming math into physical sensation. It’s math you can literally hear. One of the most incredible examples is FM synthesis.

FM stands for frequency modulation. That means that you change the frequency of a note over time, generally in a cyclical pattern. Our ears interpret frequency as pitch, so this may sound like a pitch that fluctuates over time. However, if the fluctuation occurs fast enough, our ears can no longer track it, and it begins to sound completely different.

Try listening to these demos:

Demo 1

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Origami: Aperiodic Chevron Tessellation

Aperiodic Chevron Tessellation

Aperiodic Chevron Tessellation, designed by me

Did you hear?  Someone discovered an aperiodic monotile!  Obviously, these are origami life goals.  And, I’m making it out like a joke, but I’m pretty sure I’m not the only origamist who was thinking that.

Oh, but this origami isn’t the aperiodic monotile.  Instead, I read their paper, and was inspired to create a different aperiodic tiling.  And in the mean time, I learned how an aperiodic tile ticks.

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The Ant and the Universe

In my time as a puzzle enthusiast, one of the puzzles I encountered was called the ant and the rubber band. It was only later that I realized that this puzzle had some cosmic significance.

Problem Statement

We have an ant that is trying to crawl from one end of a rubber band to the other. But as the ant crawls, the rubber band also stretches out. The ant crawls one centimeter per second. The rubber band starts out one meter long, and stretches out one meter per second. This is one of those magical math rubber bands that can stretch indefinitely. Let’s just say the ant is mathemagical too. Will the ant ever reach the end?

At first glance, it looks bad for the ant. The ant crawls crawls one centimeter closer, but falls a whole meter back. So the ant is losing about 99 cm per second. That doesn’t sound like a path to victory.

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Good old puzzles

While I’m on the subject of bad puzzles from 1995, I want to briefly share one of my favorite good puzzles from the time. Around 1995, I received a book titled 100 Perceptual Puzzles by Pierre Berloquin. It’s apparently a newer edition of an older book titled 100 Geometric Games, copyright 1976.

The book contains a wide variety of puzzles, mostly of the sort that rely on pictures, or require you to draw pictures. Many people are familiar with the puzzle where you have a 3×3 grid of dots, and you’re asked to draw four straight lines through all the points without lifting your pencil. That puzzle is not in this book, and instead it includes multiple harder versions!

Other puzzles include: mazes, spot the difference, match moving puzzles, shape counting puzzles, and knot puzzles. The knot puzzles! I will share one knot puzzle.

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Risk neutrality in EA

Effective Altruism (EA) is a community focused on donating money to create the greatest good in the world. This is mostly (?) unobjectionable–but there’s problems. The EA community has a number of philosophical viewpoints that most external observers would consider absurd, and which materially affect their donating behavior.

In particular, many people in EA believe that the most efficient way to create the greatest good is by preventing extinction caused by AI. EA surveys suggest that about 18% of community members donate to “Long term & AI” causes, compared to 62% that donate for global health & development. Clearly concern about AI is not a unanimous viewpoint in EA, but you have to imagine the kind of community where everyone takes it seriously.

EA has been under the spotlight in current news because Sam Bankman-Fried–recently arrested for massive fraud at his cryptocurrency exchange FTX–was a vocal proponent of EA. More than a vocal proponent, he founded the FTX Future Fund, which committed $160M in charitable grants to various EA causes. At the top of Future Fund’s cause list? AI.

Although I’m critical of EA, I actually think it’s a bit unfair to pretend that they’re directly responsible for SBF’s fraudulent behavior. Instead I wanted to focus on some of SBF’s philosophical views, which are shared by at least some parts of the EA community. Specifically, let’s talk about the idea that charitable causes are risk-neutral.

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