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Making Full Use of Three Ordinary Dice

The fact that rolling three dice gives a total the distribution of which approximates a bell curve has long made it the method of choice for calculating random player attributes in fantasy role-playing games.

Three plain, apparently identical, dice are more versatile than this, because even when the dice are not distinguished by being in three different colors, there are still many possible combinations that can be rolled.

The following chart:

is an attempt to make it very easy to quickly look up any one of those 56 possible combinations.

It begins with a column on the left for the 6 rare combinations in which all three dice show the same number, 1-1-1, 2-2-2, 3-3-3, 4-4-4, 5-5-5, and 6-6-6. The background of the space on the chart corresponding to that combination has one of six colors, which represent each of the six faces of the die.

In the middle section, the 30 combinations where two dice show the same number, and another die shows a different number, are charted. Here, one first looks for the combination by the number which forms the pair, and then proceeds along the diagonal, color-coded by that number, to find the odd value. Again, this is not particularly difficult to do.

In the third section, the original contribution of this diagram is found. When all three dice show different numbers, how can the chart be organized so that a particular combination can be found at a glance?

The third section begins with the four combinations 1-2-3, 2-3-4, 3-4-5, and 4-5-6, in which the dice show three numbers in sequence, since these four combinations, corresponding to the Poker hand of a "straight", are the most noticeable.

When three dice show three different numbers, the color coding is based on the number in the combination that is neither the highest nor the lowest, the number that, in sequence, is in the middle.

Then, one starts from the combination with three numbers in sequence having that same middle number. If the lowest number of the three has to be decreased by one, proceed one step upwards and to the right. If the highest number of the three has to be increased by one, proceed one step downwards and to the right.

Also, note that this section of the chart is organized so that the first column, containing combinations with three numbers in sequence, has combinations where the smallest and largest number differ by two; the second column contains the combinations where the smallest and largest number differ by three; the third column contains the combinations where the smallest and largest number differ by four, and thus are either 1 and 5 or 2 and 6; and finally the fourth column contains the combinations in which the smallest number is 1, and the largest number is 6, the only case where they can differ by five.

In order to also have all the combinations in rows corresponding to their totals, this has led to the combinations linked to 2-3-4 and those linked to 3-4-5 becoming intertwined; hence, color coding is used in the diagram.

Each dice roll has, below it, on the left, four numbers, corresponding to rolls of d7, d8, d9, and d10. A roll with three numbers the same is ignored, as shown by the dashes, for d7 and d10; here, only 210 of the possible 216 combinations are used.

On the lower right, there is a number from 1 to 36, corresponding to a d36 roll; these numbers have been assigned according to the Poker ranking of the dice rolls. This can be converted downwards in a number of ways:

d36  d6 d12  d36  d6 d12  d36  d6 d12  d36  d6 d12  d36  d6 d12  d36  d6 d12
-----------  -----------  -----------  -----------  -----------  -----------
  1 | 1   1    7 | 1   7   13 | 1   1   19 | 1   7   25 | 1   1   31 | 1   7
  2 | 2   2    8 | 2   8   14 | 2   2   20 | 2   8   26 | 2   2   32 | 2   8
  3 | 3   3    9 | 3   9   15 | 3   3   21 | 3   9   27 | 3   3   33 | 3   9
  4 | 4   4   10 | 4  10   16 | 4   4   22 | 4  10   28 | 4   4   34 | 4  10
  5 | 5   5   11 | 5  11   17 | 5   5   23 | 5  11   29 | 5   5   35 | 5  11
  6 | 6   6   12 | 6  12   18 | 6   6   24 | 6  12   30 | 6   6   36 | 6  12

although it seems odd to convert the roll of three dice down to d6, when that can be obtained just by just rolling one of them.

On the upper right, there is a number from 0 to 100; these numbers are arranged so as to be uniformly spaced, but as there are only 56 possible rolls, not all combinations are possible. The use of these numbers will be explained later.

Efficient Digit Production: d1000 in Two Rolls

Following the number for the d10 interpretation of the roll, there is a letter, either A, B, or C on the one hand, or either X or Y on the other. These letters are for the purpose of allowing a three-digit number to be generated with only two rolls of three indistinguishable six-sided dice.

There are 216 possible combinations when three dice are rolled. When a digit from 0 through 9 is being produced, the 6 combinations where all three dice show the same number are ignored, leaving 210 combinations, so each digit has 21 combinations that represent it. Of those 21 combinations, each of the letters A, B, and C each correspond to three combinations that cannot be otherwise distinguished, and the letters X and Y each correspond to six combinations that cannot be otherwise distinguished.

The charts on the lower right show how the additional information from the letter following the digit might be used to determine one other thing from two rolls of three dice, in addition to two decimal digits.

Since the letters correspond to seven groups of three combinations, where two pairs of groups are combined into one indistinguishable unit, from two rolls, we have 49 possibilities to divide up, with blocks of 1, 2, or 4 possibilities to work with.

The first chart shows how we can easily and efficiently use the letters from two rolls to obtain a number from 1 through 7, which we could use with the digits obtained to produce a number from 100 through 799.

Of course, that isn't typically what we might want. Instead, a number from 000 to 999 might be desired. The second chart shows how we can represent each of the 10 digits with four possibilities out of the 49, leaving nine possibilities over, shown in yellow. One could also have five possibilities for each digit from 1 through 9 as well, by leaving out the combinations for zero, shown in light blue-green.

But it is possible to be more efficient than that. The problem is that there are 49 possibilities instead of 50. Any pair of digits from 00 through 99 is represented by 21 times 21 combinations in two rolls of three dice, but these combinations are grouped into blocks of 9, 18, and 36 combinations which are not distinguishable, and the difficulty is that we therefore have 49 blocks of 9 combinations to divide into 10 parts.

However, the number of possibilities involving the rolls of triples, where the same number comes up on all three dice, is in excess of the 900 additional combinations which we need. Thus, the possibilities shown in yellow can be used, if we include 9 extra combinations for each digit which will represent an additional way of obtaining the numbers from 000 through 999. This is done here by only counting some of the combinations for each digit which involve a pair on two of the three dice, either followed or preceded by a triple, as shown in the table which follows the second chart.

Thus, there are 216 * 216 or 46,656 combinations of the rolls of three dice.

If the 210 combinations that indicate a digit from 0 to 9 are the only ones used, that leaves 44,100 combinations.

The second chart shows how the 441 combinations for each pair of digits directly determined by the two rolls can indicate a digit from 0 through 9. In the section colored yellow, 9 combinations are given to each digit from 1 through 9. In the rest of the chart, 36 combinations are given to each digit from 0 through 9.

The possibility for 0 is marked in blue-green, so that it could be excluded to provide a direct way to choose one of 900 possibilities, and if instead 1,000 possibilities are desired, the area marked in yellow would have to be omitted.

But the table following the second chart shows how all the possibilities in the second chart can be kept in, by adding 900 possibilities that would otherwise have been ignored, where one of the two rolls is a triple. So instead of using only 44,100 out of 46,656 combinations, now 45,000 combinations are used.

In order for this to work as shown, the digit shown in the second chart will need to be taken as the most significant of the three digits generated, followed by the digit directly generated by the first roll, then the digit directly generated by the second roll.

The contents of that chart will be repeated here:

  A B C X Y
A 1 2 3 9 4
B 4 5 6 7 2
C 7 8 9 5 0
X 0 2 4 3 8
Y 5 7 9 1 6

and for the examples below, the letter for the first roll will select the row in the chart, and that for the second roll will select the column.

The chart then shows the additional combinations used. If the first roll is any triple, and the second roll is one associated with the letters A, B, or C (thus, a roll involving a pair), we have an additional combination with the most significant digit being 0, the second digit being a digit from 1 through 6, selected by the triple, and the third digit being that directly generated by the second roll.

If, instead, the second roll is the triple, then only the triples 1-1-1, 2-2-2, 3-3-3, and 6-6-6 are counted, and the first roll, again, must be a pair. The most significant digit is zero. The second digit is 7, 8, 9, or 0. The third digit is the one directly indicated by the first roll.


1-1-2 3-6-6   0A 7A   107
1-1-5 2-4-4   3A 6B   236
2-2-6 1-5-5   9A 0C   390
2-2-3 2-3-5   6A 8X   968
2-5-5 2-3-4   1A 7Y   417
2-2-4 1-3-5   7B 5Y   275
4-4-4 3-4-4   -- 7C   047
3-3-5 3-3-3   3B --   093

An All-Purpose 3d6 System

In some role-playing games, characters have various attributes, such as Strength, Intelligence, Constitution, Dexterity, and possibly Wisdom and even Charisma. Sometimes, when a new character is created, these characteristics are determined by the total of rolling three ordinary six-sided dice.

A compact notation has come into use with role-playing games to refer to different combinations of dice. A number to be produced by the roll of one die with numbers on it from 1 to 20 is termed d20; a number to be produced by the total of the faces on two dice with numbers on them from 1 to 8 is termed 2d8.

Thus, it might be in a role-playing game that one kind of sword does 2d8 damage when it strikes, and another sword does only 2d6 damage.

A pair of 10-sided dice, where a roll of two zeroes counts as 100, can be used to produce an effective d100. Originally, icosahedral dice with the digits from 0 through 9 repeated twice were used; now, the trend is to use icosahedral dice only as d20, and dice combining two near-pyramidal regions (the faces are kite-shaped instead of triangular to permit an offset of half a face) are used as d10 or in pairs for d100.

Simplified game systems, where the d100, or the d20, or a d30 (based on a solid with diamond-shaped faces, corresponding to the edges of either a dodecahedron or an icosahedron) used for almost all purposes are currently common, as opposed to those requiring a complete set of dice corresponding to the Platonic solids (d4: tetrahedron, d6: cube, d8: octahedron, d12: dodecahedron, d20: icosahedron) are becoming increasingly common.

For ease of calculation, while full resolution to 100 possibilities is not provided, the numbers in the upper right-hand corner of the spaces above for combinations of three dice are given with a range from 0 to 100.

The combinations of three identical numbers, which each occur only once, are assigned to the values 0, 20, 40, 60, 80, and 100. These combinations can also be used, if desired, to indicate a critical hit for success, or a fumble for failure.

The remaining 50 visibly distinct dice rolls consist of 20 combinations where all three faces are different, which each occur 6 times, and 30 combinations where there is a pair of like faces, which each occur 3 times. For uniformity, then, they are spaced like this:

1 pair
3 mixed
5 pair
7 mixed
9 pair

The case where two pair combinations follow one another does not have closer spacing, except when a three-of-a-kind combination falls between. Thus, although a pair should only cover half the space covered by a mixed combination, alternating them assigns a reasonable value to each one.

In the form of a table, it would be:

111   0
222  20
333  40
444  60
555  80
666 100

112   1    122   9    133  21    144  39    155  59    166  75
113   5    223  19    233  31    244  49    255  65    266  85
114  11    224  29    334  45    344  55    355  71    366  89
115  15    225  35    335  51    445  69    455  81    455  95
116  25    226  41    336  61    446  79    556  91    566  99

123   3    234  29    345  67    456  97
124   7    134  13    235  37    245  53    346  83    356  93
125  17    135  27    145  43    236  57    246  73    256  87
126  33    136  47    146  63    156  77

The totals from 3 to 18, to produce a number on a bell curve from 1 to 100, could be interpreted according to this table:

 3   13
 4   18
 5   23
 6   28
 7   33
 8   38
 9   43
10   48
11   53
12   58
13   63
14   68
15   73
16   78
17   83
18   88

which gives symmetry around 50 and 51 being in the middle.

However, the approximate d100 numbers produced by three dice are constructed to be symmetric on a scale of 0 to 100. This can be brought into harmony by adding one to all results below 50.

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