Design Patterns: Triangular Scoring

A quick note before we get into today’s post. I recently appeared on the fantastic Ludology podcast, hosted by Geoff Engelstein and Gil Hova, where I talked about this series of design patterns. Some of you may be reading this now because you heard that episode. Welcome! If you’re a regular and want to listen, you can get the episode right here. Thanks again to the hosts for having me for a very enjoyable discussion. Let’s get on with the content!

Second note: I’ve made some corrections as pointed out in the comments. Thank you to everyone who helped clarify and correct!

For some people, the numbers are 4, 8, 15, 16, 23 and 42. But for game designers, they’re 1,3,6,10,15 and 21. That series, which can be extended out indefinitely, is called triangular because you can illustrate these numbers as a triangle.

Just by looking at them, you can see the pattern – the distance from one number to the next is always one greater than the gap between the current number and the previous one. The gap from 0 to 1 is 1. The gap from 1 to the next number will be 2, making the next number in the series 3. The gap from 1 to 3 is 2, so the gap from 3 to the next number will be 3, making that next number 6. And so on! The actual formula for finding the nth number in the series is n(n+1)/2.

Triangular numbers are everywhere in games. Hare & Tortoise, Sushi Go!, Castles of Burgundy, and my own Seikatsu, are just a small sampling of the games that feature triangular numbers. The sequence itself can sometimes be modified by adding or subtracting some constant. For example, 1,2,4,7,11,16 simply adds 1 to each number in the series (starting from 0).

Why is triangular scoring so prevalent? What is it that makes this pattern so effective relative to other possible patterns?

It’s probably worth mentioning that having scoring (or any kind of meter, not necessarily just victory points) be set to a pattern has many advantages,both for designers and players. The predictability helps players learn the game, and the patterns tend to make balancing easier by at least providing a framework.

The simplest pattern is a linear one – 1 wood gets you 1 VP, 2 wood get you 2 VP, etc. But designers will frequently need to use some kind of escalating pattern in order to make mechanisms like set-collection and engine-building work. These patterns allow the game to provide rewards or outputs that increase more rapidly than the rate of the input.

That’s all straightforward enough, but triangular numbers have some special features that make them especially suitable for game design. Like all escalating sets, triangular numbers reward players for specialization (or, if they are used to price commodities, punish players for over-exploitation). However, the shape of the curve triangular series has some important features.

Consider another common series, the powers of 2, as compared to triangular:

2^X = 1, 2, 4,  8,  16, 32, 64
Tri    = 1, 3, 6, 10, 15, 21, 28

The difference here is that the triangular series provides greater returns in the early part of the set relative to the powers of 2. But as we get to the 5th term or greater, the exponential series provides greater returns, and by the 7th term, that difference is enormous and only grows larger.

(Note: though I presented the powers of 2 as starting at 1 point, that would actually be a set of zero cards, for example, since 2^0=1. Games that use the powers of 2 will usually provide 1 point for the first card, so the more accurate notation would be 2^x-1. My apologies for any confusion, and thanks to everyone in the notes who helped point out math issues).

In a game, this means that exponentially scoring incentivizes narrow specialization. There’s no point in collecting only 2-3 items from a couple of exponentially scoring sets, as they score so poorly. Inevitably, the winner of a game with this kind of scoring, absent some other balancing or scoring mechanisms, will be the person who maxes out a single set. (This is why 7 Wonders provides set-scoring for the three science types in addition to quadratic scoring of for collecting one type. The alternate set scoring both creates incentives for collecting any type of science cards. By creating greater demand for all the science types, it is harder for any one player to specialize in one type and blow out their science scoring. If not for this mechanism, players would either routinely pass science cards not of their favored type, or be forced to hate-draft science cards they didn’t want, for few points, to prevent their neighbor from running off with the game.)

Triangular scoring, on the other hand, provides solid returns for smaller sets. Whereas the powers of 2 set values the first two cards equally (each is worth 1 point), in a triangular set the second card is worth twice what the first card was worth. While specialization is still quite valuable, smaller sets are relatively speaking more viable. Powers of 2 only becomes significantly more valuable when you collect the 6th member of a set, so collecting diverse sets can be a better strategy than narrow specialization.

Another important consideration is that triangular scoring can be inverted into triangular costing, as in Hare & Tortoise. In that classic race game that won the first SdJ award in 1979, players pay a triangular cost in carrots to move forward. Moving forward 1 space costs 1 carrot, but moving forward 3 spaces costs 6 carrots, and moving forward 6 spaces costs 21 carrots. As you can see, the cost per space climbs from 1 to 2 to 3, and so forth. Escalating costs make moving forward in big chunks inefficient in terms of carrots, but moving forward in efficient small steps is inefficient in terms of turns. This pattern can also model things like exhaustion of natural resource (ie it takes more workers/time/money to extract gold from deeper and deeper into a mine, or oil from a well, etc.) Exponential costs, by comparison, can be too cheap in the beginning, and then suddenly utterly prohibitive.

One interesting example of triangular costs is Amun-Re’s bidding system. In Amun-Re, you’re simultaneously bidding on many lots, and each lot has fixed bid amounts. The first legal bid on any lot is 1, then 3, then 6, etc. When you’re out-bid on a given lot, you must re-bid on some other lot – you can’t simply raise back. The triangular bidding helps to shorten the total auction time by forcing you to bid in ever-increasing numbers. If you wish to bid higher than the next valid bid you may, but the triangular increase means that if you want to go up two spots from a bid of 3, you now need to bid 10!

Triangular scoring is also very easy for players to work with when calculating the incremental value of a play. When you’re trying to decide whether to take your 3rd gear in 7 Wonders, you know you’ll score 9 points for the set. But the real question is how much MORE are you going to score relative to having only 2 cards. The answer is the score for a set of 3, minus the score for a set of 2, or 9-4=5. The value of that 3rd card is 5 incremental VPs.

How about the value of your 3rd dumpling in Sushi Go? It’s triangular, so the answer is easy: it’s 3 incremental VP. You can check the math – a 2-card set is worth 3 points, a 3-card set is worth 6 points. 6-3=3. But players don’t need to do the math, or even remember the sequence. The incremental value of the 4th card is 4 points, and the 5th card is 5 points. The incremental value of the nth card in a triangular set is n. I sometimes wonder whether games should list triangular number scoring charts as +1,+2,+3,+4 instead of 1, 3, 6, 10 to make this more explicit. I know from teaching a lot of Seikatsu that when players realize that this is how the flower scoring works their eyes light up with delight.

Triangular numbers are just one of the many possible sequences that designers have at their disposal, but their unique shape of high on the low end but lower at the high end make them particularly suitable to boardgames and the size of sets that we typically work with in this medium. What are some of the most interesting and innovative uses you’ve seen of this sequence? Let me know in the comments!

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