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# Some Further Possibilities

Three other possible ways to interpret the roll of three dice are shown in a modified version of the chart:

as I feel that this image does make it particularly easy to find a particular roll of three dice.

Below each roll of three dice, we see again the four numbers which represent the equivalent of that roll if the three dice were used to replace a d7, d8, d9, and d10. Then, a fifth number, giving the number that can be used if the three dice are to replace a d12, is given. That number, instead of being written in black, is either blue or red, so that one can add 12 to all the red numbers, and get a number which allows three dice to replace a d24.

Up above, in the previous chart, we had numbers in the top right corner which allowed the three dice to behave as an imperfect d100, where not all the numbers were possible, or equal in probability, on the basis that while dice producing a uniform scale of results might be useful for role-playing games, the precision of percentile dice was not really necessary. In the lower right of this chart, a different alternative is shown.

As another approach to the finest possible uniform scale of numbers that might be obtained from the roll of three dice, a number is produced that is scaled to the roll of a d60. The 20 rolls where all three dice show different numbers give the precision of a d20, but the number that is shown is a d20 roll multiplied by 3, so the numbers 3, 6, 12... 60 are given to those 20 rolls.

And then the 30 rolls with a pair are scaled to the same expanse; since there are 30 of them, the numbers from 1 through 30 that a d30 would give are multiplied by 2, so the numbers 2, 4, 6... 60 are given to them.

And the 6 rolls with three numbers the same obtain the values 10, 20, 30, 40, 50, and 60.

Thus, if some number from 1 to 60 represents the bar for the success of an action, while the type of combination of the three dice will change how finely that probability is reflected, for any of those possibilities, the scale of numbers will be uniform.

Of course, it might make sense to subtract 5 from each of the multiples of 10, and 1 from each of the multiples of 2, so that those uniform scales would be centered around the same place as well. But then, where would one put the multiples of 3?

In the upper right hand corner, then, is a set of numbers allocated so that the roll of three dice can be interpreted, again imperfectly, as the roll of a d60, but with a reasonably uniform scale of possible results from 0 to 60 (rather than from 1 to 60).

Finally, in the middle of the right side of each square, is a number from 0 to 100 once again, but this time the arrangement is not intended to produce a linear scale.

Instead, the largest and smallest values are indicated by the rolls of three dice the same, which can be made in only one way. Then, the next larger and next smaller values are indicated by rolls of three dice which include a pair, and which can be made in three ways. The values centered around the average value of 50 are assigned to rolls of three dice where all three dice are different, these arrangements being capable of being made in six ways each.

Since 101 numbers, rather than 56, are being assigned, there are unused values, and they are assigned so as to compensate, but only partially, for the different probability of the different types of combinations.

For the combinations that can be made only one way, the number assigned increments by one each time, for a density of one. For the combinations that can be made three ways, the number assigned increments by one and one-half on average, for a density of two. For the combinations that can be made six ways, the number assigned increments by two and one-half on average, for a density of two and two-fifths.

This kind of scale could perhaps be referred to as being suitable for an extent roll; the values given by the dice are stretched out over a long range, but still have a vestige of the character of an approximation to the normal distribution that the total of three dice has. Or, the scale largely behaves like a linear scale, but is tapered off at the edges to allow the less common combinations to be used for a finer discrimination of odds.

### To d216 Through Flexibility

It's true that green and red dice are quite common, and so one can have six-sided dice of three different colors by just going to the dollar store instead of going to the comic book, hobby, or game store.

Generally speaking, in an RPG, often one is using dice not to determine a particular number from all the alternatives, but simply to roll a value that is either less than or greater than a specific target number.

In that case, if one gives up assigning the combinations of the dice to specific values in a fixed order, three identical dice are perfectly capable of performing all the pass-fail tests with the exact same probabilities as could be performed by three different dice.

There are twenty combinations where the three dice are all different; each of them can be made in six ways, so this is a total of 120 combinations out of the 216 possibilities. This is more than half, which presents a slight problem, as these combinations are somewhat harder to keep track of than the others.

However, we can split them into the four "straights", 1-2-3, 2-3-4, 3-4-5, and 4-5-6, and all the others. The four straights represent 24 combinations, and all the others represent 96 combinations.

What one has left are the six combinations, each made in only one way, of three dice the same, 1-1-1, 2-2-2, 3-3-3, 4-4-4, 5-5-5, and 6-6-6, and thirty combinations, each made in three ways, of one pair. These can be ordered like this: 1-1-2, 1-1-3, 1-1-4... 4-6-6, 5-6-6, ordering first by the pair, and then by the remaining number.

If we take an arbitrary number from 1 to 216, and we want to know whether we have rolled that number, or have rolled under or over it, then we can proceed as follows:

• Let X start out as being the same as our target number.
• If X is 97 or greater, count the roll of a combination with all three numbers different, other than a "straight", as under the target, and then subtract 96 from X for our further approportionment of the combinations. Otherwise, count the roll of a combination with all three numbers different, other than a "straight", as over the target.
• If X is 25 or greater, count the roll of a "straight" as under the target, and then subtract 24 from X; otherwise, count the roll of a "straight" as over the target.
• Divide X by three, keeping the quotient, and replacing X by the remainder. Place the quotient in Y. If X is zero, subtract one from Y, and set X equal to three.
• Count off, starting from 1-1-2, as many of the thirty combinations including a pair as are indicated by the number in Y. These combinations will be counted as under the target, and all the other combinations in this group will be counted as over the target. If Y is equal to thirty-one, count all the combinations with a pair as under the target, and add three to X.
• If X is 1, then 1-1-1 will equal the target, and 2-2-2 and the other triples will be over the target; if X is 2, then 1-1-1 will be under the target, 2-2-2 will be over the target, and 3-3-3 and the other triples will be over the target, and so on.

Another way to understand this, which is rather simpler than the specific algorithm given above, is the use of the following table:

```1-1-1: 1     1-1-2:  +3   1-3-3: +33   1-5-5: +63
2-2-2: 2     1-1-3:  +6   2-3-3: +36   2-5-5: +66
3-3-3: 3     1-1-4:  +9   3-3-4: +39   3-5-5: +69
4-4-4: 4     1-1-5: +12   3-3-5: +42   4-5-5: +72
5-5-5: 5     1-1-6: +15   3-3-6: +45   5-5-6: +75
6-6-6: 6
1-2-2: +18   1-4-4: +48   1-6-6: +78
2-2-3: +21   2-4-4: +51   2-6-6: +81
2-2-4: +24   3-4-4: +54   3-6-6: +84
2-2-5: +27   4-4-5: +57   4-6-6: +87
2-2-6: +30   4-4-6: +60   5-6-6: +90

1-2-3, 2-3-4, 3-4-5, 4-5-6: +24
all others: +96
```

It is desired to make one of the six rolls of all three numbers the same correspond to a roll of exactly the target number. Initially, they correspond to the numbers from 1 through 6.

If the "straights" are counted as below the target, this adds 24 to the meaning of the triplet rolls; if the other combinations of three different numbers are counted as below the target, this adds 96 to the meaning of the triplet rolls.

To move the meaning of the triplet rolls to precisely the right spot, the chart of the rolls which include a pair is used. The number next to a combination with a pair is what is added to the value of the triplet rolls when that combination, and all the other pair combinations below it in order, are counted as below the target.

Note that this method allows more than one way to allocate the rolls for most given targets, and this has options not reachable by the algorithm given previously.

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