It's a well-known risk of game design that in pursuit of originality, you will bump into well-studied concepts of mathematics. Some game designers come straight out of maths and sciences, so they already have a library of statistical and numerical knowledge at their disposal. The rest of us come out of a creative field where... let's just say math and logic wasn't a primary focus.
All that to say, I stumbled into the term "Triangular numbers" a while back while figuring out the scoring for Koi Pond's ribbons. In layman's terms – that is to say, my terms – triangular numbers are a sequence of numbers that increase at a predictable rate. In the example below, you can see the typical progression of 1, 3, 6, 10, 15, and so on
Ticket to Ride's scoring progression was quite clearly arranged with the balance and logic of some mysterious underlying structure. I knew it was there, but I didn't know it had a name, let alone a whole history of research. They're "figurate numbers," starting with linear numbers, then triangulars, squares, tetrahedrals, and more.
- Linear: 1, 2, 3, 4, 5, 6, 7, 8, 9...
- Triangular: 1, 3, 6, 10, 15, 21, 28, 36, 45...
- Square: 1, 4, 9, 16, 25, 36, 49, 64, 81...
- Tetrahedral: 1, 4, 10, 20, 35, 56, 84, 120, 165...
Many thanks to W. Eric Martin for actually telling me what this stuff was called. I also listened to a classic GameTek segment from Geoff Engelstein covers triangular numbers in all sorts of games. This GeekList features several games using triangular numbers, including Coloretto, Amun Re, Hare and Tortoise and more. There's also an Online Encyclopedia of Integer Sequences, that catalogs many many of these particular sequences of numbers.
So, I wanted to pitch a few ideas I had for using these number sequences in games.
#1: Triangular Values for Linear Sets
There are some obvious uses in set collection mechanics. It's common in many games that a set of two matching items would be more than double the value of one item and that three of those items would be worth more than triple the value of one item. For sake of examples to follow, let's call these red beans. In this scheme, you'd score 1 point for 1 red bean, 3 points for 2 red beans, 6 points for 3 red beans, and so on. If you went with this scheme in your own game, you'd have to make forming sets part of the tactical and strategic challenge.
#2: Linear Values for Triangular Sets
Alternately, you could make sets very easy to create, but force players to get increasingly larger quantities in order to claim higher rewards. For example, you'd score 1 point for 1 red bean, 2 points for 3 red beans, 3 points for 6 red beans, 4 points for 10 red beans, and so on. Yes, these are actually diminishing returns for each step up the ladder, but there are ways to make this an interesting play experience. See Bohnanza as a key example, which uses set limits, hand limits, and assymetrical trading to make the struggle for one more card very exciting.
#3: Triangular and Square Values for Linear Sets
Now things get really weird. Let's continue with the "red bean" example and expand that to other foods and colors, like "red apple," "yellow apple, "green apple," "red bean, "yellow bean," "green bean. Now we have a 3x2 grid of attributes, with each item in the game being either red, yellow, or green and either a bean or apple. In play, the value of a set of cards could have two cumulative values, one based on color and the other based on food. In this case, let's say you use "food" to determine the triangular value and "color" to determine the squared value: 1 point for 1 red, 4 points for 2 reds, 9 points for 3 reds, and so on, plus 1 point for 1 bean, 3 points for 2 beans, 6 points for 3 beans. Thus, a set of 3 red beans would be worth 15 points. Forming a perfectly matched set is very valuable indeed!
#4: Linear and Square Values for Linear Sets
Say you wanted to reward diversity and hegemony in your set collection game. The simplest way would simply be to use the Triangular-for-Linear scheme first described above for matched sets. Then you add a twist: for each set of one-of-a-each-kind, you gain a flat rate of around 8 points. This makes each diversity set more valuable than a set of three matches, but not quite as valuable as a set of four matches. When players reach that critical decision point, they must decide whether to pursue either of those two paths to victory.
#5: Linear, Triangular, and Square Values for Triangular Sets
Returning to our second example, let's say your game requires increasingly larger sets to gain just one more point of value. However, each component of that set had three attributes. Let's say besides just "color" and "food" there was also some other attribute, like "age," such that ripe foods were more valuable than seeds. Thus you have three divergent methods of scoring a set. Say "color" has linear value for triangular sets (1:1, 2:3, 3:6, 4:10...), "food" has triangular value for triangular sets (1:1, 3:3, 6:6, 10:10), and "age" has square value for triangular sets (1:1, 4:3, 9:6, 16:10). Now that would be a brain-burning set-collection game!
#6: Beyond Victory Points
So far, these examples focus on direct conversion of sets to victory points, but you can just as easily use a linear, triangular or square sequence for other resources. Imagine a civ game in which rare resources were earned at slower rates. For example, assuming triangular sets, the rarest goods would be earned at a linear rate, the moderate goods at a triangular rate, and most common goods at a square rate. That's the beginning of a whole economy right there. Why not set up an auction where each bid must be an increasingly higher triangular number?
I'm going to explore these last two ideas more thoroughly in future set collection games, perhaps with an eye towards keeping it intuitive and easy to comprehend mid-game. I wouldn't want players being stricken with total analysis paralysis while trying to navigate three simultaneous point trackers.