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Natural Numbers In Game Design

In maths, ‘natural numbers’ are the ones you might use to count observable, whole things: eg. there are six people here. Anything that doesn’t work in place of ‘six’ there, like 3.4 or -2, is not natural. They’re kind of ‘numbers you can see’.

I’d like to use the term in game design to mean specifically that: numbers you can see. Things that are represented so simply and wholly and countably that you don’t need to display an actual numeric figure to tell the player how much they’re seeing. They can just see.

Here’s an FTL screenshot, showing how it presents your ship’s various weapons and systems:

FTL Panel

FTL is a game with masses of systems and numbers and configurations, which are part of what makes it great: there’s a huge range of possible ship loadouts to put together, and a huge number of sophisticated ways for them to go wrong. But I never would have played it, and I suspect it wouldn’t have got popular, if it felt complex. It doesn’t – you learn it pretty quickly, and pretty much anyone can play.

For me, that was down to how ‘natural’ the numbers involved were, especially those power pips shown above. Do I have enough power to bring my drone systems online? You can see the three green pips of spare power on the left, and the two empty pips next to the drone system on the right, and you know immediately: yes. Interestingly, I don’t think this even involves any counting or comparing of numbers, in the mathsy sense – for small values like this, we have some kind of low-level spatial recognition routines that just come back with an instant “yes”.

It’s further simplified by how these natural numbers reuse their same values in all contexts: the hitpoints of each system is equal to the power it’s consuming and that’s also the total power of subsystems it can support. These are concepts you have to learn, but learning them is easy because there’s no maths to do, everything is one-to-one.

Here’s a screenshot of Heat Signature:

Heat Signature Approach

The player, in the small ship, is approaching a big ship. These are randomly generated and have lots of properties that vary from ship to ship, all of which are important. But there’s no stats panel for them, here or anywhere else, showing that info as bars or figures. Partly inspired by FTL, I show everything I can in ‘natural’ numbers.

What’s this ship’s attack rating? Well, it’s got one gun. That, to me, goes one better than answering the question. If an info panel said “Attack rating: 1”, you still don’t know what that means in real terms. I’ve spent hundreds of hours in RPGs without ever finding out what their ‘attack’ stat actually means – sometimes it’s like accuracy, sometimes damage, sometimes attack speed. But if you’ve ever seen a gun fire in Heat Signature, you know everything there is to know about this ship’s offensive capabilities right off the bat.

What’s its manoeuvrability rating? If you’ve ever seen a thruster fire in Heat Signature, you know that it has 4 of them and you have a sense for how it’ll feel to fly.

How many hits can it take? Well, one hit destroys one module, so just look at how many modules it has. Also inspired by FTL, things have a one-to-one-to-one relationship wherever possible. One gun fires one missile. One missile destroys one room. One gun takes up one room. So if you’ve ever seen one ship shoot another, you’ve already permanently memorised every conversion factor for every relationship between attack and defence, because ‘one’ is as natural as numbers get. So you can see at a glance how many hits this ship can take.

You probably can’t count how many, at a glance. But what I like about natural numbers, if you use them throughout, is that you don’t need to. There’s no reason you need to know whether there’s 13 modules here or 15, because nothing else is expressed in terms of figures. You can see that there’s that many.

That visual, spatial, intuitive sense data is more immediate, more memorable, and easier to compare than a dry number. If you see a ship with more guns than this has modules, no number comparison is necessary to see what happens next – in fact, you can’t help but picture it.

And your sense of by how much that ship is outmatched is more powerful than if you’d learnt it from figures: if you were on the smaller ship, the urgency to get off it would be that much more intense.

Echo Nolan: The concept is well taken, but please don't invent new terminology that overlaps with precise mathematical terms. The value of the phrase "natural number" is that it describe the concept fairly precisely. I know confusing terminology is super common, but there's no need to make the situation worse.

Taha Nasir: A cool exploration, something maybe I should consider for my current game.

I realize your probably extremely busy and won't see this, but I have a question or two:

I wonder how one uses GM to create a potentially infinite 'room'? Do you just create a room for space, and make its width and height like, 999999? (Obviously you don't, but I'm a moron and can't figure it out)

I also wonder regarding your level of interactivity at EGX. Finally, I'll be able to check out HeatSig and talk to you, but will you be open to conversation and stuff? Because I know some devs just want to talk about their game, but I remember the guys who made Monstrum last year were super open to random conversation about Dota and stuff.

Looking forward to finally playing this tomorrow!

Jason L: Yup, I've been thinking about this for the last few years. There are always exceptions but in general the more digits there are in any game's numbers the less I think of the design.

Steve C: This might not be the same but I prefer games that use lower number values as a standard or form of progression.

For example, I think it's more simple to recognize or digest the value of "5" or "50" rather than "5,000." A lot of JRPGs tend to have higher values when it comes to stats and I find them to be less satisfying.

Cletus Van Damme: The "low-level spatial recognition routines" are called subitizing, and work for amounts up to four. I guess studies show it's a completely different action in the brain than counting.

Ressud: Definitely agree that calling it natural numbers is a bad idea. Had me really confused there for a moment. I do agree about the concept as a whole though, and find it to be pretty similar to the "show, don't tell" of writing.

Aidan: Is this something you're going to roll out to the gun variables, like spread and noise? I seem to remember seeing gameplay footage that indicated they are currently small numbers like 2.5.

Urthman: Yeah, it should be "Whole Numbers" because some space ships have zero guns.

Taha Nasir: Ah, crap. I don't know where I got the impression you were coming this year, though I coulda sworn it. You didn't miss anything; by far the worst EGX so far. Didn't play a single AAA game, though how big a loss that is is uncertain.

But yes, beginning to get a grasp on this 'natural numbers concept'. Still an interesting, thought-provoking article.

Praetor: Actually, natural numbers can include zero, depending on who you talk to (just Googled it). In this case I think it's fine as we are talking about that ways humans deal with 'natural', easy-to grasp, quantifiable things.

Jim: There is a great video explaining how the human brain analyses numbers and proportion on Vsauce's youtube channel. Basically if you are shown some objects, let's say chairs, then you can instantly count them if there are 1, 2, 3 or 4 chairs. It's called subitizing and everyone can do it without conscious thought or analysis. Above this though and your brain finds it harder and has to start estimating, or deliberately counting. This is why when we use tally marks for counting, we make four vertical strokes before making one horizontal stroke to indicate a tally of five. If we kept making vertical strokes our brains would start to struggle. The Vsauce video is here:

https://www.youtube.com/watch?v=Pxb5lSPLy9c

There is an online counting test he discusses which effectively demonstrates how difficult it can be for your brain to estimate and compare larger numbers of objects here:

http://panamath.org/... ...phase1.php

rpgdan: This is a great rule thumb.

A lot of visual information gets handled subconsciously, we know there are four guns, without having to use our executive brain function to consciously count them. It's information we get for free. The the executive function is free to deal with more interesting problems in the gameworld.

rpgdan: Oh yes like Jim has already written (far more precisely and clearly than I did :D ). The video link looks cool - I'll check it out.

Magnificentophat: @Taha Nasir. I think Tom explained it in one of his heatsig dev logs. The room's infinite because the player stays in one place and everything moves around them and the room is only as big as the screen, but has no bounding box or whatever.

Jason L: I've been thinking about this post (along with Making A Game To Test A Critique) the last couple weeks as I've surprised myself with my level of excitement for 2D fighting game Pocket Rumble. I'd originally backed its Kickstarter because of its design around simplified inputs; its low hit point count and 1-hit-1-damage system were both ideas I'd been kicking around in my head for years, but I considered them subordinate to the elimination of obstructive input challenges. Actually that's still true, the design would still work with traditional HP counts, but I think its use of low natural numbers has had a catalytic effect on my perception of the game by making footage of it both dramatic and immediately comprehensible. EVO Moment 37 notwithstanding, the "hero pixel" is normally a tedious accounting artifact and rarely produces that effect.

It's a bit amusing that humans' capacity to subitize is apparently the same as that famously measured in rabbits. Hrair chairs.