As you know, I’m a bit of a math nerd. When I first started playing D&D, I was annoyed at the progressions that were used for experience tables, and various other things. Progressions were simply a series of linear segments. For instance, experience would increase by 1000 for a few levels then by 1500 for a few levels, then by 2000 for a few levels. This served the purpose but it wasn’t elegant. As a math nerd, I wanted it to be elegant as well.

### Searching For a Solution

I played with endless equations that generated the type of smooth curving slope I wanted, within the ranges I wanted. However, everything I came up with was forced and obscure. I wanted a simple equation but all the simple equations I tried just didn’t “feel” right. Then I got lucky.

### Found It!

I made a list in one column, numbered from 1 to 20. In the second column, on each line, I wrote the sum of the numbers in the first column down to that line. In the third column, on each line, I wrote the sum of the numbers in the second column down to that line. Immediately, I knew that this family of functions was something special. It is simple, elegant, and exactly what I wanted. More than that, the special relationship between the various functions within the family would make possible an unexpected benefit. But I’ll get back to that later.

### The Equations

It isn’t convenient to create the series manually. Even using a spreadsheet it is tedious so I set to determine the formula for each series. The list of numbers from 1 to 20 doesn’t require a formula but it helps to see the pattern in the other formulas if you represent it as (n+0)/1. The first sequence is then represented as (n+0)/1 * (n+1)/2. The second sequence is represented as (n+0)/1 * (n+1)/2 * (n+2)/3, and so on. Obviously, you would replace (n+0)/1 with n. I list it in the other form because I think it helps to see the relationship.

### Triangular Numbers

Obviously, I wasn’t the first person to discover these functions. However, I asked a number of math professors and no one recognized them. Beyond that, I didn’t really make much attempt to research them though.

Recently, I was doing some research on dice probability and discovered a site that allows you to search for a function based on the series generated by the function. Fantastic! I entered the first few numbers generated by the first function and was overjoyed with the results. This sequence is referred to as the Triangular Numbers. There was all sorts of information on uses and references for more information.

### Pyramidal Numbers

Excitedly, I entered the first few numbers from the second sequence and hit search. The second series is referred to as the Pyramidal Numbers. There was all sorts of info on this one as well.

### Unexpected Benefit

Earlier I mentioned that the special relationship between the functions gave me an exciting and unexpected benefit. I actually discussed this in an earlier post (Smoothing Out The Experience Tables) but I will summarize it again here.

I use the first function (times 10) to determine monster experience with n = level. I then use the second function (times 100) to determine the experience required for each level of advancement. If you have ten balanced encounters you receive ten times the monster experience. This exactly matches the experience required to go from the corresponding level to the next higher level. The relationship between the two functions allows this to work out perfectly without having to manhandle the progression in order to make it work out. This is shown in more detail in the earlier post on the subject.

### Why Is This Important

It isn’t. The original progressions used in various editions of D&D work just fine. The latest edition, matches up monster experience and the experience tables so that ten encounters gains a level (more or less). These sequences don’t provide any earth-shattering benefit.

But they LOOK good. They are elegant. They feel right. And they make math nerds like me feel all warm and fuzzy just looking at them.