About Dice Rings

Over on the Falkirk RPG website, there was a brief debate about "Dice Rings". These, as the name implies, are rings that can serve as dice - they have a fixed inner ring and a loose outer ring marked with numbers. To roll a random number, you spin the outer ring, and when it comes to a halt there's a read-out point at which you get your number.

My contention is that these dice rings are a neat gimmick, but that I would very much doubt the randomness of the dice. In particular, it is my contention that the result is coupled to the initial state of the ring.

Now, it was noted on the original kickstarter that the dice have been extensively tested for randomness. And this website provides more details about their testing.

Firstly, to ensure that the dice can't be 'aimed' or otherwise cheated, the numbers around the ring have been placed in a carefully-selected order. Where possible, the numbers alternate between odd and even values and between high and low results. And while '1' and '20' aren't directly opposite one another, they are close to being opposites.

Secondly, the creator tested the dice by spinning them over and over, recording the results, and checking there were no anomalies - that each result came up as often as any other.

So that's it, right?

Actually, no.

Consider this: suppose your PC is having a hard time of it. You've got three attacks per round, but you need a nat '20' to hit. You've just rolled one such '20', and are now rolling your second attack (or rolling to confirm the crit). Either way, you spin the dice...

Now, in this scenario, it doesn't matter that the dice have been checked to ensure that, on average, they roll each number as often as any other. And it also doesn't matter that you have a 50% chance of an 11 or more, or a 50/50 chance of an odd or even number.

What matters is that on this roll, the odds of scoring that natural '20' need to be exactly 5%. No more, no less.

Obviously, the order of the numbers around the ring does nothing for this - whatever order is chosen you'll still need the ring to spin through exactly N revolutions, where N is an integer. But what about the testing described in "secondly", above?

Well, here's the thing: the test that he describes is effectively a "Chi Square" test, which can indeed be used to test a die for randomness. The problem is, though, that many pseudo-random generators would also pass such a test. (Indeed, if you had a generator that simply stepped from '1' to '2' to '3', up to '20' and then to '1', it would still pass that test. Although in that case, the uniformity of the results would itself be a warning sign.) To be sure the outcome wasn't tied to the initial state, you would need to run this test 20 times - once for each of the initial states of the ring.

My expected behaviour of the spin ring is as follows: each person will operate slightly differently, but each will naturally adopt a favoured technique for spinning the ring. This technique will cause the outer part to spin through X revolutions (where X is not necessarily an integer), plus or minus Y.

Now, Y may very well be more than a full revolution of the ring. So, that's good enough, isn't it?

Again, I'm afraid the answer is "no". Because the distribution of values won't be even - like virtually everything else in the sphere of human endeavour, it will sit on a bell curve - the most likely single outcome will be for a spin of exactly X reolutions, then the next most likely will be one position to either side, and then the next position outwards, and so on.

This means that, depending on the initial position of the ring, the outcomes will themselves have a bell curve distribution, not the even distribution that is required. The most likely outcomes will vary, but will depend on the initial state of the die.

And, for that PC who needs a natural '20', the odds of actually scoring that value are unlikely to be exactly 5%. Sorry, your character's dead.

Now, all that said - three more things.

Firstly, as the designer notes, these "spin rings" are based on something called "worry rings", which are sometimes suggested as a means to cure anxiety - the wearer fiddles with them mindlessly, in order to distract his mind from his worries. I would expect spin rings to work extremely well for such a person - by constantly fiddling, they'll effectively be pre-randomising the starting position of the ring, and so give a random aspect (much like a well shuffled deck of cards). They would, however, need to do this every time between rolls, or they'll still never confirm that crit. (Conversely, it's likely the rings would work especially badly for someone such as myself, who tends towards reducing entropy by regimenting my dice.)

Secondly, it is by no means guaranteed that any particular die is properly random anyway. Indeed, depending on how much faith you put in the Gamescience presentation, it appears likely that most dice aren't properly random. In which case, little is lost by switching to spin rings - you're moving from one imperfect method to another.

Thirdly... if you really want to use a spin ring in favour of a d20 in one of my games, I won't stop you. I find the topic interesting, but mostly for the theoretical exercise in reasoning. When it comes to actual practice, I really don't care all that much.