Inspired by https://gurpsland.no-ip.org/articles/PolyGURPS.htm, which gets rid of ±3 modifiers, but still preserves ±2, expands the result range, and also doesn't go far enough to make use of Catalan dice.
Implications of Central Limit Theorem must be upheld. Singular die has uniform distibution, 2 dice have a triangual distribution, 3 dice start to look like Bell curve, and the more dice, the more normal the distribution, and the smaller the spread.
NOTATION CONVENTIONS:
Constant: d1
Coin: d2
Platonic: d4, d6, dF, d8, d12, d20
Trapezohedra: đ10, đ10*10 (maybe also d14, d16, d18, d22, d26, d28, and d32)
Catalan: d24, d30, d48, d60, d120
Weird: d100 (equivalent to d% pseudodie)
đ means either use a 0-based dice or just read the die face count as 0. Conversely, d means either use a 1-based dice or just read the 0 as the die face count. đ% means read 00 as 0, and d% means read 00 as 100. dF rolls -1, 0, or 1.
Odd-dice are typically even-dice with each number twice.
d48 and d120 may be a bit harder to obtain in Eastern Europe due to international shipping and customs. In case of diffulties with obtaining other dice as well, factorize:
100 = 2 * 2 * 5 * 5 => đ10 * 10 + d10
60 = 2 * 2 * 3 * 5 => đ 6 * 10 + d10 (or đ30 * 2 + d2)
48 = 2 * 2 * 2 * 2 * 3 => đ 8 * 6 + d 6 (or đ12 * 4 + d4)
30 = 2 * 3 * 5 => đ 6 * 5 + ceil(d10/2)
24 = 2 * 2 * 2 * 3 => đ 6 * 4 + d 4 (roll hour as d12 with d2 for AM/PM)
20 = 2 * 2 * 5 => đ10 * 2 + d 2
12 = 2 * 2 * 3 => đ 6 * 2 + d 2
8 = 2 * 2 * 2 => đ 4 * 2 + d 2
4 = 2 * 2 => đ 2 * 2 + d 2
Cannot do anything about lack of d2. You may be able to do something about lack of d10 and d6 if you somehow have some weirdly shaped d5 or d3. Luckily, d10 comes twice a D&D set (ignore or make up tens as you go), d6 is available in normie stores, and for d2 pay for dice sets in cash and use change.
NPC ROLLS (DirectDice):
All Nd6-N rolls can be DirectDiced as Nđ6, but this is boring. Each plain 2d6 are also substituted for d8+d4. If these run out, try d10+d2.
Reject means to reroll on said result. If the extrema of both sides are mentioned, it's possible to treat it as a slight range extension.
d6-1 0 5 đ6
d6 1 6 d6
d6+1 2 7 d8 1, 8
d6+2 3 8 d8 1, 2
d6+3 4 9 đ10 0, 1, 2, 3
2d6-2 0 10 đ8+đ2
2d6-1 1 11 đ10+d2
2d6 2 12 d8+d4
2d6+1 3 13 d8+đ6 2
2d6+2 4 14 d8+d6 2, 3
2d6+3 5 15 4d4 4, 16
3d6-3 0 15 đ8+đ6+đ4
3d6-2 1 16 đ8+d6+đ4
3d6-1 2 17 đ8+d6+d4
3d6 3 18 d8+d6+d4
3d6+1 4 19 đ10+d6+d4 2, 3
3d6+2 5 20 5d4
3d6+3 6 21 5d4+d2 22
4d6-4 0 20 3đ8 21
4d6-3 1 21 3đ8 0
4d6-2 2 22 2d12 23, 24
4d6-1 3 23 3d8 24
4d6 4 24 2d8+2d4
4d6+1 5 25 3d8+d2 26
4d6+2 6 26 3d8+d2 5
4d6+3 7 27 7d4 28
5d6-5 0 25 2đ12+đ4
5d6-4 1 26 2đ12+d4
5d6-3 2 27 2d12+đ4
5d6-2 3 28 2d12+d4
5d6-1 4 29 3d10 3, 30
5d6 5 30 2d8+d6+2d4
5d6+1 6 31 2d8+4d4 32
5d6+2 7 32 2d8+4d4 6
5d6+3 8 33 2d8+4d4+d2 7, 34
Generally, to ensure minimum result, roll that many dice, and to ensure maximum, ensure the faces add up to it. At medium amounts of d6, the minimum starts getting too high for this approach.
HEROIC ROLLS (AltDice):
These balance matching mean and range, like in the original table. Replacement is by single type of dice. Precise numerical meanings of results of such rolls are unlikely. Shift the corrective constant accordingly.
3d6+3 6 21 3d8 +3
4d6-3 1 21 2d10 -1
4d6-1 3 23 2d12 +1
4d6+4 8 28 4d8 +4
5d6-1 4 29 3d10 +1
5d6+2 7 37 3d12 +3
6d6 6 36 2d20 +4
6d6+1 7 37 4d10 +3
7d6-2 5 40 5d8 0
7d6+3 10 45 5d10 +5
8d6-2 6 46 4d12 +2
At higher amounts of d6, too many polyhedral dice are rolled, albeit less, facing the same problem of counting taking too long, as well as the range being expanded too much. You need to be quite a dice hoarder to have 16 d8 or 12 d12, which only come once per D&D set unlike d10. As a hotfix, use 2d8=d10+d6=d12+d4 and 2d12=d20+d4.
SUPERHEROIC ROLLS (LessDice):
Hero System and Shadowrun are notorious for using obscene amounts of d6, moreso than GURPS. They're hard to hold in hands, and the pips take ages to count. This is the time to finally resign at eliminating corrections. 4 or 5 dice (6 or 5 from 54d6, 6 or 7 dice from 63d6, 8 or 7 from 80d6) are enough to invoke Central Limit Theorem, allow the Bell curve to take shape, and provide enough wiggle room for mean coinage payment problem. These mostly Catalan dice aren't typically owned in large quantities of the same type, the most egregious ask is for 3d48 (substitute with d100+d24+d20) and 3d30 (substitute with d48+d30+d12). 3d120 enters only at 61d6 and 3d100 at 63d6.
Mean 1.5 2.5 3.5 4.5 5.5 6.5 10.5 12.5 15.5 24.5 30.5 50.5 60.5
Note that 2d6 have the mean of d14.
18d6 18 63 108 d48+d30+d24+d20
19d6 19 66.5 114 2d30+2d24+d20
20d6 20 70 120 2d48+2d20
21d6 21 73.5 126 d48+d30+d24+2d20
22d6 22 77 132 2d48+d30+d24
23d6 23 80.5 138 d48+2d30+2d24
24d6 24 84 144 3d48+d20
25d6 25 87.5 150 2d48+d30+d24+d20
26d6 26 91 156 d60+2d48+d24
27d6 27 94.5 162 3d48+2d20
28d6 28 98 168 2d60+d48+d24
29d6 29 101.5 174 3d48+d30+d24
30d6 30 105 180 d120+d48+d30+d8
31d6 31 108.5 186 d100+d48+d30+d24+d10
32d6 32 112 192 d100+2d48+d24
33d6 33 115.5 198 d100+d48+d30+2d24
34d6 34 119 204 d120+d60+d48+d6
35d6 35 122.5 210 d100+d60+d48+d20+d12
36d6 36 126 216 d120+d60+d48+d20
Eventually, the amounts of Catalan dice needed start rising as well, and the variance becomes enormeous, so a different approach is needed. Consider splitting digits, like rolling 6d6*10+4d6 instead of 64d6. See NPC Rolls table for 3d6 to 5d6 equivalents. Don't roll 1d6 or 2d6 for most significant digit, Bell curve starts only taking shape at 3d6. Numerical inflation makes a TTRPGs feel like Yu-Gi-Oh!, where players become accountants with 8000 LP, instead of more pedestrian 20 Life of MTG.
DEMIGOD ROLLS (NormalDice):
At higher amounts of d6, the Bell curve gets so narrow it's almost better to roll a bunch of dF and modify the mean, so for example 100d6 goes from 100 to 600 with mean 350, but instead of d120+5d100+d48+d24 (8 large dice) or 10d6*10 (digit split loses precission with round numbers, maybe try different radices), roll 350+10dF*10, which gives flatter range from 200 to 500. Generate the least significant digit with d10 if you need precission. This keeps the variance increase in check. There could be a NormalDice table that specifies the exact amount of Fudge dice to replicate the distribution, but just note this:
Nd6: Mean 7N/2 = 3.5N, Variance 35N/12 = 2.9167N, StdDev 1.7078*N^(1/2)
KdF: Mean 0, Variance 2K/3 = 0.6667K, StdDev 0.8165*K^(1/2)
LdF*10: Mean 0, Variance 200L/3 = 66.6667L, StdDev 8.165*L^(1/2)
Therefore, K = 35N/8 = 4.375N, or N = 8K/35 = 0.22857K, and L = 7N/160 = 0.04375N, or N = 160L/7 = 22.85714L
Using dF*10 instead of dF reduces the amount to be rolled to approximate normal distribution to 4 from 438 at 100d6, which is still quite some distance away from true normal, therefore to reach the range of 100 to 600, there needs to be 25 decadic dF rolled, i.e. N/4. Mean is fixed in NormalDice mode, it's just a question of whether to match the variance or the range, and if you don't know, arithmetic mean is 4+25/2=14.5, which rounds to 15. Multiply accordingly for 1000d6 or other crazy rolls.
This means the amount of dice rolled is reduced by a factor of 25 when matching variance (realistic), 4 when matching range (cinematic), and 6.667 when balancing with arithmerical mean. I propose a factor of 10, which happens to be a geometric mean of 4 and 25, and is simpler to conceptualize for decimal beings who create market for artrocities like d100, but post-graduate GMs may have their own opinions.
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