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# One Table to Rule Them All

In some role-playing games, the impacts of different types of weapons or spells are specified by designations such as 3d6 (the total of the roll of three six-sided dice, each one numbered from 1 to 6), 2d12, 4d8, and so on and so forth.

One would have a certain probability to hit or miss, modified by one's dexterity score, and if one had strength above that needed merely to wield one's weapon, one might get a damage bonus when one hits.

This involves a number of limitations.

Combinations need to be chosen from the available dice, and one would need multiple sets of dice to use several copies of the same die. Weapons even doing d8+d10+d12 damage are not common in RPGs. What the combinations are really attempting to approximate, at least in many cases, is a distribution with a given mean and standard deviation.

Adding something to damage as an adjustment is common enough, but multiplying damage by a factor is too awkward to be considered.

A much larger table could easily be made, for example based on the fact that the 53rd root of 2 allows good approximations to factors of 3 and 5 in integral powers thereof, (one might, in practice, choose to use the 176th root of 10) and one could use d100 or even d1000 to supply the input randomization, but something more practical, say based on the available values of resistors and capacitors with 5% tolerance, and perhaps using a d20 as input, ought to be sufficient.

A d20, however, has an even number of faces; what would really be liked is something with an odd number of faces, so that it would be possible to roll the exact average amount of damage. A further simplification, with the ability to extend further out to the edges of the distribution, without cluttering up the table with many close-together values, would be to use, say, 2d6 as input.

Thus, a table like this could be made the basis of a very powerful and flexible RPG system based on the ordinary 2d6, used for advancing one's token in humble and ordinary board games:

```          |  Roll (7 is always 0)
Standard  |      6     5     4     3     2 (- negative)
Deviation |      8     9    10    11    12 (+ positive)
----------+----------------------------------
100      |     39    76   115   159   220
112.2018 |     44    86   129   179   247
125,8925 |     49    96   145   201   277
141,2538 |     55   108   162   225   311
158.4893 |     62   121   182   253   349
177.8279 |     70   136   205   283   391
199.5262 |     78   153   230   318   439
223.8721 |     88   171   258   357   493
251.1886 |     99   192   289   400   553
281.8383 |    111   216   324   449   620
316.2278 |    124   242   364   504   696
354.8134 |    139   271   408   565   781
398.1072 |    156   304   458   634   876
446.6836 |    175   342   514   712   983
501.1872 |    197   383   577   799  1103
562.3413 |    221   430   647   896  1237
630.9573 |    248   483   726  1005  1388
707.9458 |    278   541   814  1128  1558
794.3282 |    312   607   914  1266  1748
891.2510 |    350   682  1025  1420  1961
1000      |    393   765  1150  1593  2201
```

So, if one has a sword which causes damage with a mean of 40 (actually 39.8107 rounded) and standard deviation of 10, but the damage is reduced by a multiplier of 0.8 (actually 0.7943282 rounded) we just need to move two steps down in the table for both items: 40 becomes 32 (31.62278 rounded), 10 becomes 8 (7.943282 rounded) and the table of damage for that sword with that multiplier contains entries derived as shown below:

```Roll  Damage Calculation  Damage
2     31.62278 - 17.48    14
3     31.62278 - 12.66    19
4     31.62278 -  9.14    22
5     31.62278 -  6.07    26
6     31.62278 -  3.12    29
7     31.62278            32
8     31.62278 +  3.12    35
9     31.62278 +  6.07    38
10     31.62278 +  9.14    41
11     31.62278 + 12.66    44
12     31.62278 + 17.48    49
```

Although it looks complicated, the intent is that adjustments which involve multiplication as well as addition can be obtained through the simple operations of moving up or down in the table, adding or subtracting, and moving the decimal point left or right.

In practice the quantities would be rounded before being added, so the results would actually be symmetrical around the rounded center point; and, since very large numbers would not normally be needed, in practice the table used would be more likely to look like this:

```          |  Roll (7 is always 0)
Standard  |      6     5     4     3     2 (- negative)
Deviation |      8     9    10    11    12 (+ positive)
----------+----------------------------------
10  |      4     8    12    16    22
11  |      4     9    13    18    25
13  |      5    10    14    20    28
14  |      6    11    16    23    31
16  |      6    12    18    25    35
18  |      7    14    20    28    39
20  |      8    15    23    32    44
22  |      9    17    26    36    49
25  |     10    19    29    40    55
28  |     11    22    32    45    62
32  |     12    24    36    50    70
35  |     14    27    41    57    78
40  |     16    30    46    63    88
45  |     18    34    51    71    98
50  |     20    38    58    80   110
56  |     22    43    65    90   124
63  |     25    48    73   101   139
71  |     28    54    81   113   156
79  |     31    61    91   127   175
89  |     35    68   103   142   196
100  |     39    76   115   159   220
```

And the table could be replaced by other tables, converting to the same set of steps in standard deviation from rolls of another chosen set of dice instead of the totals of 2d6, but this one table could be used to calculate all the various types of adjustments used in an RPG.

If one wishes to use a d20 instead of 2d6, the table, which, being now based on a dice roll with a linear scale, may illustrate the principle at work more clearly, becomes:

```          |  Roll
Standard  |     10     9     8     7     6     5     4     3     2     1 (- negative)
Deviation |     11    12    13    14    15    16    17    18    19    20 (+ positive)
----------+----------------------------------------------------------------
100      |      6    19    32    45    60    76    93   115   144   196
112.2018 |      7    21    36    51    67    85   105   129   162   220
125.8925 |      8    24    40    57    75    95   118   145   181   247
141.2538 |      9    27    45    64    84   107   132   162   203   277
158.4893 |     10    30    51    72    95   120   148   182   228   311
177.8279 |     11    34    57    81   106   134   166   205   256   349
199.5262 |     13    38    64    91   119   151   186   230   287   391
223.8721 |     14    42    71   102   134   169   209   258   322   439
251.1886 |     16    48    80   114   150   190   235   289   362   492
281.8383 |     18    53    90   128   168   213   263   324   406   552
316.2278 |     20    60   101   143   189   239   296   364   455   620
354.8134 |     22    67   113   161   212   268   332   408   511   695
398.1072 |     25    75   127   181   238   301   372   458   573   780
446.6836 |     28    84   142   203   267   337   417   514   643   876
501.1872 |     31    95   160   227   300   379   468   577   721   982
562.3413 |     35   106   179   255   336   425   526   647   810  1102
630.9573 |     40   119   201   286   377   477   590   726   908  1237
707.9458 |     44   134   226   321   423   535   662   814  1019  1388
794.3282 |     50   150   253   360   475   600   742   914  1143  1557
891.2510 |     56   169   284   404   533   673   833  1025  1283  1747
1000      |     63   189   319   454   598   755   935  1150  1440  1960
```

And, if you really want to get fancy, you can double the number of multiplicative steps, and roll 4d6, which leads to the following table:

```          |  Roll (14 is always 0)
Standard  |     13    12    11    10     9     8     7     6     5     4 (- negative)
Deviation |     15    16    17    18    19    20    21    22    23    24 (+ positive)
----------+----------------------------------------------------------------
100      |     28    56    84   114   144   174   207   242   283   337
105.9254 |     30    59    90   120   152   185   219   257   300   357
112.2018 |     31    63    95   127   161   196   232   272   318   378
118.8502 |     33    67   100   135   171   207   246   288   337   400
125.8925 |     35    71   106   143   181   220   260   305   357   424
133.3521 |     37    75   113   151   191   233   276   323   378   449
141.2537 |     40    79   119   160   203   246   292   342   400   476
149.6236 |     42    84   126   170   215   261   310   362   424   504
158.4893 |     44    89   134   180   227   277   328   384   449   534
167.8804 |     47    94   142   191   241   293   347   407   476   565
177.8279 |     50   100   150   202   255   310   368   431   504   599
188.3649 |     53   106   159   214   270   329   390   456   534   634
199.5262 |     56   112   169   227   286   348   413   483   565   672
211.3489 |     59   119   179   240   303   369   437   512   599   711
223.8721 |     63   126   189   254   321   391   463   542   634   754
237.1373 |     66   133   200   269   340   414   491   574   672   798
251.1886 |     70   141   212   285   361   438   520   608   711   846
266.0724 |     75   149   225   302   382   464   550   645   754   896
281.8382 |     79   158   238   320   404   492   583   683   798   949
298.5381 |     84   167   252   339   428   521   618   723   846  1005
316.2276 |     89   177   267   359   454   552   654   766   896  1065
334.9652 |     94   188   283   380   481   584   693   811   949  1128
354.8132 |     99   199   300   403   509   619   734   859  1005  1194
375.8372 |    105   211   318   427   539   656   778   910  1065  1265
398.1069 |    112   223   336   452   571   695   824   964  1128  1340
421.6962 |    118   237   356   479   605   736   872  1021  1194  1420
446.6833 |    125   251   377   507   641   779   924  1082  1265  1504
473.1509 |    133   265   400   537   679   826   979  1146  1340  1593
501.1868 |    140   281   423   569   719   874  1037  1214  1420  1687
530.8840 |    149   298   449   603   762   926  1098  1286  1504  1787
562.3408 |    158   315   475   639   807   981  1163  1362  1593  1893
595.6616 |    167   334   503   676   855  1039  1232  1443  1687  2005
630.9567 |    177   354   533   716   906  1101  1305  1528  1787  2124
668.3433 |    187   375   565   759   959  1166  1383  1619  1893  2250
707.9451 |    198   397   598   804  1016  1235  1465  1715  2005  2383
749.8934 |    210   421   634   852  1076  1308  1551  1816  2124  2524
794.3274 |    223   446   671   902  1140  1386  1643  1924  2250  2674
841.3942 |    236   472   711   955  1208  1468  1741  2038  2383  2832
891.2499 |    250   500   753  1012  1279  1555  1844  2159  2524  3000
944.0598 |    265   530   798  1072  1355  1647  1953  2287  2674  3178
1000      |    280   561   845  1136  1435  1745  2069  2422  2832  3366
```

Or, scaled down, and more practical, as above, but still using 4d6, the table could be:

```          |  Roll (14 is always 0)
Standard  |     13    12    11    10     9     8     7     6     5     4 (- negative)
Deviation |     15    16    17    18    19    20    21    22    23    24 (+ positive)
----------+----------------------------------------------------------------
1      |      0     1     1     1     1     2     2     2     3     3
1.1  1 |      0     1     1     1     2     2     2     3     3     4
1.3  1 |      0     1     1     1     2     2     3     3     4     4
1.4  1 |      0     1     1     2     2     2     3     3     4     5
1.6  2 |      0     1     1     2     2     3     3     4     4     5
1.8  2 |      0     1     2     2     3     3     4     4     5     6
2    2 |      1     1     2     2     3     3     4     5     6     7
2.2  2 |      1     1     2     3     3     4     5     5     6     8
2.5  3 |      1     1     2     3     4     4     5     6     7     8
2.8  3 |      1     2     2     3     4     5     6     7     8     9
3.2  3 |      1     2     3     4     5     6     7     8     9    11
3.5  4 |      1     2     3     4     5     6     7     9    10    12
4    4 |      1     2     3     5     6     7     8    10    11    13
4.5  4 |      1     3     4     5     6     8     9    11    13    15
5    5 |      1     3     4     6     7     9    10    12    14    17
5.6  6 |      2     3     5     6     8    10    12    14    16    19
6.3  6 |      2     4     5     7     9    11    13    15    18    21
7.1  7 |      2     4     6     8    10    12    15    17    20    24
7.9  8 |      2     4     7     9    11    14    16    19    22    27
8.9  9 |      2     5     8    10    13    16    18    22    25    30
10      |      3     6     8    11    14    17    21    24    28    34
11      |      3     6     9    13    16    20    23    27    32    38
13      |      4     7    11    14    18    22    26    30    36    42
14      |      4     8    12    16    20    25    29    34    40    48
16      |      4     9    13    18    23    28    33    38    45    53
18      |      5    10    15    20    26    31    37    43    50    60
20      |      6    11    17    23    29    35    41    48    57    67
22      |      6    13    19    25    32    39    46    54    63    75
25      |      7    14    21    29    36    44    52    61    71    85
28      |      8    16    24    32    40    49    58    68    80    95
32      |      9    18    27    36    45    55    65    77    90   106
35      |     10    20    30    40    51    62    73    86   100   119
40      |     11    22    34    45    57    69    82    96   113   134
45      |     13    25    38    51    64    78    92   108   127   150
50      |     14    28    42    57    72    87   104   121   142   169
56      |     16    32    48    64    81    98   116   136   159   189
63      |     18    35    53    72    91   110   131   153   179   212
71      |     20    40    60    80   102   124   146   171   201   238
79      |     22    45    67    90   114   139   164   192   225   267
89      |     25    50    75   101   128   156   184   216   252   300
100      |     28    56    84   114   144   174   207   242   283   337
```

Even with this table, often one would be dividing by 10 and rounding to use it for calculations, so this time while the steps are as in the earlier tables, it is longer than the others because it has been extended downwards.

In the lower extension, the standard deviation rounded to tenths has been shown to allow rows to be indicated unambiguously; it is also then shown rounded to units so that that scale can be used to multiply the means.

From the table, it is easy to see that a roll of 4d6 approximates a distribution with a mean of 14 and a standard deviation close to 3.2.

So far, we've seen how to use dice to approximate a distribution with a certain mean and a certain standard deviation.

If one is working with an existing FRP game, however, such things as the damage done by a given weapon are not expressed in those terms, they're expressed in types of dice rolls.

So we need another table telling us which table to use.

One die gives a uniform distribution, and two dice give us a triangular distribution; this is true with any identical dice, not just the d6.

It's only with three or more dice that an approximation to the bell curve of the normal distribution is obtained.

So this table shows the mean and standard deviation for several common dice combinations:

```  3d4       4d4       5d4       6d4       7d4       8d4
7.5      10        12.5      15        17.5      20
1.9365    2.2361    2.5       2.7386    2.958     3.1623

3d6       4d6       5d6       6d6       7d6       8d6
10.5      14        17.5      21        24.5      28
2.958     3.4156    3.8188    4.1833    4.5184    4.8305

3d8       4d8       5d8       6d8       7d8       8d8
13.5      18        22.5      27        31.5      36
3.9686    4.5826    5.1235    5.6125    6.0622    6.4807

3d10      4d10      5d10      6d10      7d10      8d10
16.5      22        27.5      33        38.5      44
4.9749    5.7446    6.4226    7.0356    7.5993    8.124

3d12      4d12      5d12      6d12      7d12      8d12
19.5      26        32.5      39        45.5      52
5.9791    6.9041    7.719     8.4558    9.1333    9.7639

3d20      4d20      5d20      6d20      7d20      8d20
31.5      42        52.5      63        73.5      84
9.9875   11.5326   12.8938   14.1244   15.2561   16.3095
```

In the case of the d10, since the faces are marked with the digits 0 to 9 instead of the numbers 1 to 10, the corrected means are shown, where 0 is read as 10; the uncorrected means would be the same as for the d8.

Dice with 34 sides have been made because the totals from three such dice vary from 3 to 102. Subtract 2, and the range is from 1 to 100; and so this makes a bell curve distribution that matches well with the linear numbers from percentile dice in much the same way that the totals from 3d6 work well with those from a d20.

But 9d12 give totals that vary from 9 to 108 - subtract 9, one gets 1 to 100; 11d10 give totals that vary from 11 to 110 - subtract 10, one gets 1 to 100. Why are exotic dice with 34 sides needed?

In the case of 3d34, the standard deviation is 16.9926; in the case of 9d12, it is 10.3562, and in the case of 11d10, it is 9.5263.

So the distributions are very different; with nine or eleven dice, as opposed to three, the range of possible totals includes extremely unlikely events very distant from the center of the bell curve, with the likely totals all closer to the mean. Three dice produce an approximate bell curve more appropriate for an FRP game.

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