If you are a mathematician the words I use below are bound to be formally inaccurate, sorry! If you are sane, the below content is more than likely to be extraordinarily unnecessary, and probably useless, except perhaps in the most specific kinds of circumstances, sorry!
Thank you anydice.com for making this lunacy possible (<– please consider donating to this great resource).
I did this thinking, and I decided to post it here in case anyone ever wonders about the same, at some point, and finds this post, however unlikely that might be …
Never to be deterred by a hopeless cause, I began to wonder about D6 bell curves using lots of D6s.
I think this began because I supposed that dice pool games using large dice pools sum all of these D6s, but upon reflection, I suspect that is not the case.
Anyway … I began to think that the result of summing lots of D6s would be boring because it’s bound to return mostly the same number, i.e. the peak area of the curve. So, how to add more ‘swing’ to the curve, but to retain the same overall range?
To cut a long story short, there are sort of “harmonics”, where you can (i) roll the same number of dice and (ii) the overall result range is the same, but (iii) where different groups of dice are rolled … these different sets give different probability profiles.
|(i) Homogeneous D6 dice set||(ii) ‘Harmonic’ equivalent set||Result range:|
|2D6||D8+D4||both give 2-12 and use 2 dice|
|3D6||D10+2D4||both give 3-18 and use 3 dice|
|4D6||D12+3D4||both give 4-24 and use 4 dice|
|8D6||D20+7D4||both give 8-48 and use 8 dice|
The rub here is that using split dice (i.e. “harmonic” dice sets) give bell curves with more swing than a regular homogenous dice sets. The bigger the gap between the dice sizes in the non-homogeneous dice, the more swing there is. Below compares (i) D6 sets and (ii) their ‘harmonic’ dice set equivalents and (iii) overlapping the ends of (i) and (ii) for ease of comparison:
(i) Anydice: output 2d6 output 3d6 output 4d6 output 8d6
(ii) Anydice: output d8+d4 output d10+2d4 output d12+3d4 output d20+7d4
(iii) Anydice: output 2d6 output 8d6 output d8+d4 output d20+7d4
So, if you want to flatten a D6 bell curve, simply replace the number of dice with one of the non‑homogenous equivalents above. I won’t bore you with more graphs (at least for now), but take my word for it, the biggest D-number you can introduce in the ‘harmonic dice set’, the bigger the flattening effect. So, if you want to replace 8d6 with a ‘harmonic set’ equivalent you are better off using D20+7D4 rather than two sets of D12+3D4 (see table above for these ‘harmonic’ equivalents).
So, we have 2D6, 3D6, 4D6 and 8D6 covered (i.e. the rolls where there are ‘harmonic’ equivalents) – but what to do with 7D6? Good question. You could use D12+3D4 plus 3D6 (i.e. replacing the biggest harmonic set). You could even go on to replace the 3D6 part of D12+3D4 plus 3D6 with the harmonic set D10+2D4 (but this extra tweak adds little benefit).
But, there is an even stronger way to flatten the curve: use D20+7D4 and subtract D6. What we are doing here is going to the next largest ‘harmonic’ (equivalent to 8D6) – but to bring the dice range back into alignment with 7D6, we are subtracting a D6. Below shows this: the top line is 7D6, the middle line is D12+3D4+3D6 (and D12+3D4+D10+2D4) and the bottom line is using this subtraction method:
Anydice: output 7d6 output d12+3d4+3d6 output d12+3d4+d10+2d4 output d20+7d4-d6.
In fact, this ‘subtraction’ method can be used even a few steps further backwards, before it starts to give ‘wayward’ results. Here’s (i) homogeneous D6’s (ii) non-homogenous dice to replace the largest D6 set (iii) the D20+7D4 plus/minus D6s to give/restore the appropriate dice number:
(i) Anydice: output 2d6 output 3d6 output 4d6 output 5d6 output 6d6 output 7d6 output 8d6 output 9d6 output 10d6
(ii) Anydice: output d8+d4 output d10+2d4 output d12+3d4 output d12+3d4+d6 output d12+3d4+2d6 output d12+3d4+3d6 output 1d20+7d4 output 1d20+7d4+d6 output 1d20+7d4+2d6
(iii) Anydice: output d20+7d4-6d6 output d20+7d4-5d6 output d20+7d4-4d6 output d20+7d4-3d6 output d20+7d4-2d6 output d20+7d4-d6 output 1d20+7d4 output 1d20+7d4+d6 output 1d20+7d4+2d6
Basically, the D20+7D4 plus/minus D6s is the strongest way to flatten the curve, but (obviously) is quite complex, and after subtracting more than 2 or 3 D6s starts to give results that extend beyond the normal range and/or can give a negative number, i.e.:
|d20+7d4 plus/minus D6s method||In place of:||% Exceeds end ranges
(doubled to cover both ends)
|% which is Zero or less|
So you can subtract about 3D6s before the subtraction method gives an appreciable chance that a result will extend beyond the normal range, or give a number zero or less
The big picture is you can start from a bigger ‘harmonic’ set and subtract one or more D6s, as opposed to starting from a smaller harmonic and adding one or more D6s … if you can tolerate all the mathematical jiggery-pokery that is! Phew.
A final piece of craziness, sort of building on this idea – if you’ve ever wanted a D100 bell curve, well you could try 5D20-20D10+108!
Anydice: output (5d20-20d10)+108
This curve is centred between 50 and 51 and the chance of exceeding 100 or being below 0 is 0.25% each way.
Did you really read all that? Well take a Chuffty Badge you lunatic!
One day, this might be of interest to someone!