Today, I offer a new technique for rolling multiple dice many times with great efficiency. Any RPG can benefit from that!

Sometimes, the shortness of the road can make up for rougher conditions. Image by Nataly from Pixabay

I hope everyone had a wonderful Christmas break. Mine was great, though not without its challenges – but I have evidently weathered them, because here we all are, in a bright and shiny New Year!

This isn’t going to be a long post – but it is going to be a profound one. In the adventure I’m currently working on for the Zenith-3 campaign, a situation arose in which a character was going to be exposed to multiple minutes of an environment doing damage to him every turn.

Not just a few dice, but a lot of dice. Fortunately, he also has a lot of protection. How many dice, and whether or not that protection was going to be enough, would depend on what the character chose to do.

(Note that I’m being circumspect because this adventure hasn’t been run yet).

He could choose to head into the danger and incur a higher rate of damage. He could try to get out of danger by the shortest possible route – which also incurs that higher rate of damage but only for a relatively short time. Unless he gets lost along the way – a potential real danger. He has other options, as well.

So I didn’t know how many dice a round he would be taking, but I knew this: there are 3 twenty-second rounds in a minute (or 6 10-second rounds – the latter is our default, the former something I’m experimenting with). Thats 15 rolls of 8-to-10d6 every five minutes. And the character could be waiting in this situation for 20, 30, 40 minutes or more.

120 or more rolls of 8-to-10 d6 each. And apply defenses to each. And calculate damage from each. And accumulate that damage from each. And recover some of that damage from each.

It might take as little as two minutes to do each, but it would probably be more. FOUR HOURS of making rolls while everyone twiddled their fingers.

There had to be a better way. And then I thought of one, and got Google Gemini to help flesh it out and make it real.

The Principle

As you make more and more rolls, they become more and more inclined to average out. That’s one of the abiding principles harnessed by The Sixes System, and it’s something I understood very clearly. So why not leverage that fact? Roll ONCE and apply a mathematical manipulation to that result to get the outcome of R rolls.

Sounds incredibly simple, doesn’t it? Well, it’s not quite that easy, but it’s pretty close to it.

The procedure

1. Roll Once.
2. Subtract the average roll to get Delta.
3. Determine R, the number of Rolls that this calculation is going to represent.
4. Multiply the Delta by 1/ (R^0.5).
5. Add the average roll to the result.
6. Apply any modifiers that are applicable to every roll. The result is the average result over the totality of R rolls.
7. Multiply by R.
8. Apply any other adjustments. Which gives you the total of effect at the end of those R rolls.

This sounds complicated, but in most RPGs it will be even simpler.

An example

Let’s pick… 8d6 damage, 12 rolls over 12 rounds. Defenses subtract 20 from the result. Anything that gets through the defenses also does x3.5 Stun damage. At the end of each minute, the character gets 25 Body back and 50 Stun. He has a pool of 120 HP and 240 stun to draw upon.

1. I roll 8d6 and get 33.
2. The average of 8d6 is 8 x 7 / 2 = 28. Delta = +5.
3. R = 12.
4. Delta x 1 / (R^0.5) = 5 / 12^0.5 = 5 / 3.464 = 1.4434
5. Add the average roll 28 + 1.4434 = 29.4434.
6. Subtract Defenses of 20 = 9.4434.
7. Multiply by R = 12 x 9.4434 = 113.3208. Round in the character’s favor to 113. Multiply this by 3.5 for the Stun = 395 stun damage.
8. If 3 rolls is a minute, 12 rolls is 4 minutes, and the character gets 4 x 25 = 100 HP back and 4 x 50 = 200 Stun back. So his losses at the end of the 4 minutes are 113-100=13 HP and 395-200=195 stun.

That took about 5 minutes to do – but I was typing explanations. If I just did it? 2 minutes, tops – 60 to 90 seconds, more likely.

Another example

There are 25 men defending a castle wall. There are 200 archers attacking them, and each archer gets 2 shots per round. Each shot does 1d6 if it hits. The archers have a 3 in 20 chance of hitting, and half of those hits will strike castle wall instead, so it’s effectively 1.5 on d20. Archers have to inflict an 20 points of damage to kill a target.

There are a couple of preliminary calculations needed for this example.

• 200 x 2 x 1.5 / 20 = 30 hits per round.
• Distributed over 25 men, that’s effectively 1.2 hits per defender per round.
• At an average of 3.5 points per hit, that’s an average of 4.2 damage per defender per round.
• At 20 needed, that’s an average of 20 / 4.2 = 4.76 rounds of combat.

That’s all well and good, but we don’t want averages – we want specifics.

So let’s do 5 x 6d6 per round for 4 rounds and see where we’re at (5 x 6 = 30).

1. Roll 6d6.I get 18.
2. The average of 6d6 is 6 x 7 / 2 = 21. Delta is -3.
3. R = 4.
4. -3 x 1 / 4^0.05 = -3 / 2 = -1.5.
5. -1.5 + 21 = 19.5.
6. 19.5 x 4 = 78.

• 78 points distributed amongst 25 men is 3 12 points per man per round.
• For every man who’s taken twice that, there will be one who’s taken half that. So 1.56 and 6.24.*
• Repeat: 0.78 and 12.48.
• Repeat: 0.39 and 24.24.96/
• Six numbers, so out of every 6 defenders, 1 is dead, 1 is half-dead but still fighting, and 1 is wounded slightly.
• 25 defenders, so the total is 25/6=4 dead, four half-dead, four lightly wounded, 13 virtually whole.

* Assuming the roll is symmetrical.**

** Okay, this isn’t quite true – if there’s a minimum result, the true answer is half-way from the result to the minimum matches halfway from the maximum to the maximum minus the result. But this is a lot quicker and easier, and it works even when you don’t know what the maximum is, as in this case.

Specifics vs Averages – it makes a VERY big difference.

I would then run the same calculation for the defenders taking down attackers. About 4 minutes to run 4 rounds worth of siege.

But the next time around, I’d be informed by the results of the first run and increase R to 6 or 8, and run the attack in bigger ‘chunks’ of time.

Useful R values

If you can arrange it, the following R values are especially convenient, for reasons that should be obvious: 4, 9, 16, 25, 36, 49, 64. The square root of these numbers are 2, 3, 4, 5, 6, 7, and 8, respectively.

Perhaps less obvious are 2.25, 6.25, 12.25, 20.25, 30.25, 42.25 and 56.25, .These become 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, and 7.5, respectively.

Wait, What? “2.25” rolls? “2.25”” rounds? How does THAT work?

The “round” or “turn” is an artificial construct. It doesn’t actually exist, it’s just a convenient dividing line. Multiply by the number of minutes or seconds in one, and you get real-world units of, respectively, minutes or seconds.

And that works in the other direction, as well. Let’s say there are 12 seconds in a round – then 2.25 rounds is 2.25 x 12 = 27 seconds.

Or, let’s say there are 15 seconds in a round, and a character has to run through a danger zone, which will take him 72 seconds at his movement rate. 72 / 15 = 4.8 rounds. Not 4 rounds, or 5 rounds, 4.8 rounds.

Or, to go back to the original trigger for all this – the character might spend 16 minutes in the 6d6 zone, then cross 100m of 8d6, 100m of 10d6, and 200m of 12d6. Most movement rates aren’t going to translate those distances into neat time intervals when they are measured in rounds. Seconds, maybe, maybe not, but rounds? Almost certainly not.

Three Final Tips

Tip #1

If you really want your results to FEEL like you’d rolled them all, aim for an R that is one less than required and add one one totally legitimate random roll. In reality, this inflates the randomness more than is warranted, but it gives the right ‘feeling’ in play.

So if your true R is 15, use R=14. One random roll feeds into the calculation, and one stands alone. I do NOT recommend this, though – it’s an extra set of die rolls for not enough reward.

Tip #2

The second one is this: if you have a long interval, break it into smaller chunks and a smaller R, and generate a new ‘seed value’ for each chunk. For 20, 30, or 40 minutes? 5 or 6 minutes at a time. For longer? 10, or 15. For even longer? 20.

Divide the time by the total number of rolls that you want to make. That will tell you how long each chunk should be – just round to the nearest convenient number.

Tip #3

The more granular the die roll, the better this works. Let that sink in for a moment. It’s not just that the system processes 12d6 just as quickly as it does 6d6, saving more time; the results are qualitatively more nuanced.

But that granularity is also enhanced with higher R values.

That implies a sweet spot – and it’s going to be roughly found at (R x N) ^0.5. And the closer that R and N are, therefore, the closer you are to the sweet spot – without even calculating it.

if you have a choice between 15 dice and R=8 or 10 dice and R=12, the second one will give the best results.

If you have a choice between 60 dice and R=4 vs 15 dice and R=16, the second one wins every time. Not just is ease of roll, but in quality of result.

Well, that’s the power of 1 on Root R. Hopefully it’s useful out there!