Alright, nerds. Let’s talk about math. I realize this is a subject that isn’t exactly en vogue in some TRPG circles, sometimes met with “Story is more important!” or the ever popular “I’m bad at math!” Respectively, my responses to both are “It’s not a dichotomy” and “If you play D&D, you might be better at it than you realize.”
So with that said, let’s explore some of the fields and applications of mathematics as they appear in Dungeons and Dragons!
This is the simplest form of mathematics, and it covers the manipulation of numbers. Adding, subtracting, multiplication, and division are all different functions of arithmetic. Taking damage is subtracting the results of a damage die from your remaining HP. Taking half damage for a saving throw against a red dragon’s breath weapon is adding up all the damage dice rolled, and dividing that total by two.
It’s the simple stuff, and besides ascribing the name to this field of mathematics, there’s not much else of interest here. Let’s move on!
A man once told me that Algebra is a pointless complication of arithmetic, because math is supposed to be about numbers, not letters. That fool was a highschool dropout, though, and was often willfully ignorant of just about everything: the point of algebra included. The letters involved in this form of math are its defining characteristic, since they represent unknown numbers or ranges of values.
If you’ve ever wanted to figure out about how much HP a particular monster has without peeking at the Monster Manual (or other such catalogs), you likely employed algebra without realizing it. The tried and true method of counting up how much damage a creature takes before dying to ascertain its max HP is an application of Algebra, since we’re trying to figure out what MaxHP is equal to or less than.
A further application of Algebra would be using the information of how much HP a monster has to develop a strategy for fighting it, since a creature is dead if MaxHP =< Y with Y being how much damage a creature has taken. If a know a goblin has roughly 7 HP, we can practically guarantee a fireball spell will eliminate a single goblin since it does 8d6 damage (yielding a minimum of 8 damage, assuming a failed saving throw from the goblin).
“But wait, isn’t Fireball an Area of Effect Spell? Using it on a goblin is a waste!” This is true, which brings us to our next mathematical discipline!
Algebra is the foundation of several other mathematical disciplines, such as Geometry, which is mathematics as it applies to points, lines, and planes (in short, the math of shapes). To continue our Fireball and Goblin example, discovering the maximum amount of goblins one could slay with a single fireball spell would be a problem for Geometry. Dungeons and Dragons 5th edition describes fireball as a 20 foot radius sphere, centered on a single point.
D&D 5e briefly describes some of the Geometric terms it uses, but doesn’t go in as much detail as earlier editions, and how to draw out these diagrams on a grid square has lead to some lengthy discussions on how to chart these phenomena. If you read that discussion, you’ll notice that even when presented with Pathfinder’s rules regarding the geometry of spell effects, there are still numerous other viable methods that get defined!
In short, from a design perspective, it suddenly makes sense why 5th edition avoids including these lengthy sections in the books! Speaking of design, there’s one final discipline of math that any game designer worth their salt should have an understanding of.
Calculus! This is a discipline that gets a bad rap, as it’s often used synonymously with the phrase “any sort of math I’m too lazy to learn” when discussing rules-heavy games. As I mentioned earlier, though, a basic understanding of calculus is vital to game design, if not for your own games, then definitely if you’re designing new content for an already existing game.
If arithmetic is manipulating numbers, algebra is using formulae to determine unknown numbers, and geometry an application of algebra, then calculus is the next step: devising formulae to express given phenomena. Meaning if you’ve ever come up with a mathematical way to remember the numbers on a chart in D&D and it’s been accurate, you’ve used a form of calculus before.
Take for example, figuring out ability score modifiers. There’s a chart for this on page 13 of the 5th Edition Player’s Handbook (or on Roll20). You’ll notice, that the bonus for an ability score increases by 1 for every 2 that the ability score is increased by. We also know that 10-11 has an ability score modifier of 0. Observing this pattern, we can devise a formula to describe the relationship between ability score and modifiers!
(x/2) - 5 = y
In the above formula, x is the ability score, and y is the modifier. In order for this formula to work out, however, you have to round down no matter what the remaining decimal places are after you’ve divided. The computational explanation for this is that x and y for this formula are integers, and when dividing integers you discard remainders.
In other words, calculus is about noticing patterns and finding ways to define these patterns mathematically. Game design benefits from this because if we know this formula, we not only don’t need to remember the chart, but we can also find out what information would be listed on the chart!
To take this just another step further; let’s look at Proficiency Bonus and how it relates to character level. We notice that it starts at 2, and raises by 1 every 4 Character Levels. Thus, we can give it this formula:
(x/4) + 2 = y
Here, x is Character Level, y is Proficiency Bonus, and both x and y are integers (meaning we don’t deal with remainders when dividing). These are simple examples of where calculus could be applied in Dungeons and Dragons. However, if you understand the principle of noticing the patterns of how the classes are designed, you can extrapolate where they’d go further if they could go beyond level 20!
Math is just as much as part of tabletop gaming as storytelling is, and as long as we’re using numerical representations for a character’s capabilities, it always will be. This isn’t necessarily a bad thing, though. This aspect of tabletop games can even serve as an effective way to practice some mathematical concepts.
After all, you may not need Geometry in your day to day life, but if you need to know if your Elven Ranger’s Longbow can hit that dragon as it’s flying away, you’ll be glad you know how to use the Pythagorean Theorem!
If Aaron der Schaedel could find a way to use mathematics as a way to compensate for general social ineptitude, he’d find some contrived way to write an article about that, too. Until that day comes though, he’ll just stick to using TRPGs as an excuse to talk about mathematics. You can tell him to stop being a turbonerd via Twitter: @Zamubei
Picture Reference: http://www.belloflostsouls.net/2018/09/dd-combat-by-the-numbers-dungeons-and-datamining.html
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