Falling Through the Earth (and Mars)

Recently, on Reddit (of course), someone posted a video talking about how long it would take to fall straight through the center of the earth to the other side. One of the nice parts of the video is that it shows you what the time would be when we account for the actual, non-uniform density of the earth. The video just showed an Excel sheet, though.

I recommend you watch the linked video so you have a good visual of the problem. Once you’ve done that, come on back! In this post I’m going to show you how to solve the problem using Python and a numerical integrator from SciPy.

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Computational Methods for Games 2: Markov Chains

In many games, tabletop or otherwise, there are a series of positions, board states, or other features that occur in some kind of order. In monopoly, for example, you travel in a circle. Each property is a ‘state’ that your piece (battleship!) can be in. In something like Candyland, Chutes and Ladders, or Mr. Bacon’s Big Adventure, there is a goal state to reach, and you do things (roll die, draw cards, etc.) to try to get there.

What makes these more complicated than say, just figuring the combined probability of rolling a certain sum for many die, is that the game states branch. Branching just means that you can reach more than 1 state after your current one. Markov Chains are a powerful tool for analyzing a game’s progress through it states, and this post will show you an example of that, using the game Betrayal at House on the Hill.

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Computational Methods for Tabletop Games 1: Zombie Dice and Monte Carlo Simulation

Many board and tabletop games rely on the randomness of dice rolls (or card order) to create uncertainty in the game. This will lead many frustrated players to ask “What are the odds of that happening?”. Catan is a good example of this, as a run of not getting any of your numbers will quickly lead to frustration! Catan’s odds are easy to calculate, though. The die rolls are independent from one turn to the next, and the state space of outcomes never changes.

When the games get more complex, the odds may not be possible to compute analytically, or they may just be complicated to compute by hand. Zombie Dice is an example of just such a game. Zombie dice has many stages within a turn, and while each stage is analytically calculable, the overall odds of getting some number of brains is dependent on the player’s strategy. Using a general method called Monte Carlo Simulation, we can easily play thousands of turns and calculate odds, all without needing more than a simple random number generator.

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