My economic simulation of spacefaring kittens

Kittens Game is a clicker game that you can play in your browser. It makes a strong first impression, as it tempts you into choices that will kill off your kittens within twelve minutes. But I’m not here to review the game, I’m here to talk about spreadsheets!

Clicker games often support passive gameplay (e.g. leave it running overnight), active gameplay, or any combination of the above. On the very active end, you could try to optimize it, setting up spreadsheets to run calculations. So, I spent a thousand years tinkering with spreadsheets, and I liked it. There’s a story there, a mathematical story.

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The information theory of Mysterium

A question that keeps me up at night is “What is the theoretical best you can do in Mysterium?” I’m exaggerating a bit, but it is a pointless question about a silly board game that I nonetheless spent too long thinking about. I went so far as to watch a series of lectures about information theory–listening to it in the background while in dance class, as one does. I never solved the problem, but let me at least explain what the problem is.

Mysterium

Mysterium is a cooperative board game where the players are trying to solve a murder mystery via psychic communication with the victim. One player takes the role of the ghost, and the rest take the role of psychic mediums. The ghost is not allowed to speak, and may only communicate through cryptic visions. The visions are represented by cards with surreal artwork. For example, one card has two people climbing into a giant fish mouth, another has a tarantula-like thing over a chandelier. After the mediums receive their visions, they discuss what they mean and make their guesses; and the whole time the ghost giggles about how wrong they are.

Example visions: two people climbing into a fish's mouth; a polar bear and spirit owl read a book; a chandelier hanging from strings from a tarantula's mouth

Examples of vision cards.  Source: Mysterium rulebook.

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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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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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