Monday Musings: Math and Game Design

    Welcome to TwistedSpoon Studio! It's another Monday, and that means another Musing. If you're reading this blog, odds are you're into Magic; and in that case, odds are that you know that Richard Garfield, creator of Magic, has a PhD in mathematics. (Specifically, his BS was in computer mathematics and his PhD was in combinatorial math.) And you can bet that a strong understanding of math is a huge boon when it comes to designing games.

Golden Ratio by Alix Branwyn

    So, do you need a PhD in math to design games? Not at all. Mark Rosewater, the lead designer of Magic for two decades or so, studied communications and aspired to be a screenwriter. Matt Leacock, designer of the Pandemic board game, got his degree in visual communications. Gary Gygax, the father of modern gaming, only attended junior college before starting his career as a game designer.

    That being said, your game probably has numbers in it, and it's easier to work with numbers when you know a math or two. This article isn't a deep dive into discrete math or statistics or combinatorics; instead, I'm going to give you a jumping-off point and some ideas to think about.

    Alright, math is important. I guess your fourth grade teacher was right about that much. But why? Yeah, yeah, games have numbers and all that. But what about that is fun? Well, let's take a step back here. It's not just games that are math. It's art in general. And I don't just mean visual art in a golden-ratio kind of way. All art is math. Composition is all about shapes, colors, and the relationships between them. Music is about patterns in time and the relationships between notes and chords. The culinary arts are about the ratios of ingredients, flavors, and textures. Even storytelling revolves around pacing and structure. 

    The secret to creating interesting art in any form is texture. Imagine a blank page-- It's an interesting symbol, but as an art piece, it's not very engaging. That doesn't change if you make it a red page or a blue page. Homogeneity is just boring. Texture is all about making things different. In drawing, you use lines of different lengths and widths; in painting, you use different colors and shapes of different sizes; in photography, you place objects in different fields of view. Texture comes from the differing ratios between the properties of objects. The human mind is inherently keyed into ratios, whether you're aware of it or not. 

    So, how do you apply that to game design? Long story short, make things different. Short story long, you can get those numbers from anywhere. Using numbers and ratios found in life will feel realistic-- a game about pizza might have 'zas cut in four, six, eight, or twelve slices; a game about bacteria might have different types of germs growing at different rates based on their real-world reproductive cycles. (Plague Inc. is a great example of a game based on actual research; it's a great crash-course in epidemiology, if you're interested.) If your source material is something concrete, you can use the real world basis to create something engaging.

    Your other option is to engineer your game based on set specifications. For example, when Ricky G created Magic, he chose the seven-card starting hand and starting life total of 20 completely arbitrarily. The rest of the game's balance-- the value of card draw, the power of creatures, the viability of life gain-- revolves around those initial constraints. Another factor to consider is your playtime-- If you want players to score 100 points in a 10-minute card game, they need to score 10 points per minute on average. If you have four players playing in turns, that means that each one has an average of 2 minutes 30 seconds of direct playtime-- that means they need 1 point every 1.5 seconds of their play. Say turns take 30 seconds at a time; in that case, you're looking at 20 points per turn. The time-based approach varies based on your game's mechanics and execution, but you get the idea.

    Obviously, probability is probably going to play a part in your game's balance. Randomness is generally the easiest way to add variation and replayability to a game, and it makes skill-based games more approachable to new players. Whether it's cards, dice, chips, or RNJesus, it's important to understand how probability works in your game. There's enough material there for another article (or, you know, an entire branch of mathematics), but you can look to tools like this hypergeometric calculator to balance card games, or Random.Org's dice roller to test out dice mechanics. This tutorial on z-tables can help explain probability in the context of large sets. In general, though, your most powerful tool is going to be playtesting. 

    That's all for this week! I hope that was more insightful dialogue than insane diatribe, but you'll have to let me know. How do you approach balance in your own projects? How did you arrive at the numbers that you ended up using? Let us know below, or over on Reddit here.

    This Friday's FUR article is going to look at top-down designs based on animal and convention tropes; next Monday, we're going to analyze a recent Magic set and discuss how to apply that data to our own designs. Until then, check out last week's article on managing workflow here, or read the rant that inspired this series in the first place here.

See you soon!