Minecraft, the wildly popular game, features a world built entirely with cubes, which seems counterintuitive for calculating pi (π). Pi is the mathematical constant representing the ratio of a circle’s circumference to its diameter, typically requiring a perfect circle without edges or corners to be calculated accurately.
However, mathematicians Molly Lynch from Hollins University and Michael Weselcouch from Roanoke College have discovered a method to compute pi’s value of 3.14159… within the confines of the Minecraft universe.
For those not deeply familiar with Minecraft, here’s a quick overview: it’s a “sandbox” game where players can roam freely and construct a variety of structures, including buildings and circuits, using cube-shaped elements. This involves gathering and transforming resources into new materials and items.
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Minecraft’s gameplay offers extensive freedom, allowing for creativity. Past players have shown that the game is Turing complete, meaning any computer program can be executed within it. Some users have even programmed a playable version of Minecraft within the game itself!
Given this context, it’s less surprising that pi can also be calculated in Minecraft. If any computer program can be implemented, so too can one that outputs pi. However, converting an algorithm into the game world is often complex, requiring each computer instruction to be translated into a Minecraft action. This can lead to thousands of in-game instructions.
Lynch and Weselcouch aimed to avoid such complexity. Their objective was to make mathematics engaging for young audiences, using Minecraft as the medium. In a 2024 paper, they introduced several methods to compute mathematical constants like pi within the game, all with minimal complexity.
Throwing Darts at a Board
The researchers chose a straightforward method for calculating pi in Minecraft: the well-known darts technique.
Imagine playing darts poorly, like me. In this scenario, you’re aiming at a circular dartboard mounted on a square wall section. You’ll hit somewhere on the square wall but not necessarily the circular dartboard. With your lack of skill, it’s purely random whether the dart lands on the board or the wall. If you throw enough darts, you can estimate pi.
Why does this work? Suppose the square has a side length of two meters, covering an area of four square meters. The circle inside would then have a diameter of two meters, a radius of one meter, and an area of π square meters. Randomly thrown darts have a π⁄4 probability of hitting inside the circle. By counting the darts within the circle and dividing by the total darts thrown, you get a result close to π⁄4. Multiply by four to approximate pi.
In 2024, Lynch and Weselcouch applied this technique in Minecraft. They created a circular structure using red blocks with an 11-block “radius” and enclosed it within a blue square.
Next, they simulated random dart throws using Minecraft creatures called slimes. Slimes move randomly even when no players are nearby, as Lynch and Weselcouch noted. They paired these with zoglins, creatures that kill slimes.
Using these creatures, Lynch and Weselcouch generated random events that could be tracked without direct observation. They covered the red circle with funnels called hoppers, which automatically collect items that fall onto them. Every time a slime was killed, it dropped items collected by a hopper. By calculating the number of slimes killed in the circle and dividing by total kills, they approximated π⁄4.
During their game test, 619 slimes were killed, with 508 inside the circle. This data gave them an estimated pi value:
π ≈ 4 × (508 / 619) = 3.283
The authors acknowledge this isn’t a highly accurate approximation of pi. They suggest improvements like enlarging the square and circle areas and increasing the number of slimes killed. A larger circle better mimics a true circle, and the Monte Carlo method (darts technique) becomes more precise with more random events. In Minecraft, this means more slimes and zoglins.
Lynch and Weselcouch admit this method won’t ever be truly efficient. Instead, their aim is to inspire, particularly young people, by making mathematics engaging. A Minecraft battle between slimes and zoglins is likely more captivating than a highly optimized algorithm.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
