A Method for removing given results from dice rolls while preserving relative probabilities

On his internet home page, software engineer Simon Tatham has described a method for creating a set of dice that roll results with the same relative probabilities as results from two standard six-sided dice, while eliminating the possibility of running the result 7, the most common result.

In this paper, I will describe a generalization of this method to dice of other numbers of faces, and to other results than the most likely result from rolling the two separate dice.

On his website, Tatham describes his method for creating a set of dice which maintains the relative probabilities of all results for two six sided dice, except for 7, which is removed entirely from the possible outcomes.

The process first involved writing out a results table for roling two six-sideed dice and adding the results.

	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Then Tatham reorganized the results table into a different grid.

	1	2	3	4	5	6
1	7	6	6	6	6	6
2	7	8	8	8	8	8
3	7	5	9	4	4	4
4	7	5	9	10	10	10
5	7	5	9	3	11	2
6	7	5	9	3	11	12

At this point, this table could be used as a cumbersome alternative to rolling 2d6 and adding them together. However, this can be made simpler by assigning different numbers, and a series of dots, to each face of the die. When you roll the two dice, the result will be the face with a greater number of dots. In the next table, the number of dots on the face of the die will be represented by the number after the dash.

	7-6	5-4	9-4	3-2	11-2	0-0
6-5	7	6	6	6	6	6
8-5	7	8	8	8	8	8
4-3	7	5	9	4	4	4
10-3	7	5	9	10	10	10
2-1	7	5	9	3	11	2
12-1	7	5	9	3	11	12

Note now that rolling a 7 is a property of a single die, rather than both dice as with the standard rolling two six-sided dice and adding them.

By removing this face from the die, you can create a new set of two dice - one six-sided, one five-sided - that roll the results of two six-sided die additive with the same relative probabilities to each other - with the exception of the 7 result, which is not rolled at all.

	5-4	9-4	3-2	11-2	0-0
	<	<	<	<	<
6-5	6	6	6	6	6
8-5	8	8	8	8	8
4-3	5	9	4	4	4
10-3	5	9	10	10	10
2-1	5	9	3	11	2
12-1	5	9	3	11	12

It should be clear that Tatham's method as applied to two six-sided dice added together should be generalizable to two dice of any number of sides added together. Consider a set of 2 n-sided dice, numbered 1-n, with results found by adding the upward face on both dice. This would produce a results table similar to that generated by the two six-sided dice, as follows:

• The most likely result - \(n+1\) - will occur in \(n\) entries along the lower left to upper right diagonal.
• The second most likely results - \(n,n+2\) - will occur in \(n-1\) entries each, either directly above or below the entries with value \(n+1\).
• The \(i\)th most likely results - \(n+2-i , n+i\) - will occur in \(n+1-i\) entries each, either directly above or below the entries for the \(i-1\)th most likely results.
• This pattern naturally ends with \(2, 2n\) - the \(n\)th most likely results, with only one entry each in a cornerof the results table.

	1	2	…	n-1	n
1	2	3	…	n	n+1
2	3	4	…	n+1	n+2
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
n-1	n	n+1	…	2n-2	2n-1
n	n+1	n+2	…	2n-1	2n

Following Tatham's example, this results table can be reorganized, according to the following algorithm:

1. Let the leftmost column of entries of the table be filled with the most likely result of \(n+1\).
2. Let what entries remain available of the topmost two empty rows be filled with the 2nd most likely results of \(n\) and \(n+2\), with \(n\) being arbitrarily chosen as the entry to fill the higher of the two rows and \(n+2\) as the entry to fill the lower of the two rows.
3. Let what entries remain available of the 2nd and third column be filled with the 3rd most likely results of \(n-1\) and \(n+3\), with \(n-1\) being arbitrarily chosen as the entry to fill the leftmost of the two columns and \(n+3\) as the entry to fill the rightmost of the two columns.
4. Repeat steps 2 and 3 with the entries of decreasing likelihood on the available entry spaces on the results table, until all the spaces are filled and every possible result has an equal number of entries in the new results table as in the original results table.

The new results table will look something like this, assuming n is even:

	1	2	…	n-1	n
1	n+1	n	…	n	n
2	n+1	n+2	…	n+2	n+2
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
n-1	n+1	n-1	…	2n-1	2
n	n+1	n-1	…	2n-1	2n

And like this, if n is odd:

	1	2	…	n-1	n
1	n+1	n	…	n	n
2	n+1	n+2	…	n+2	n+2
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
n-1	n+1	n-1	…	2n-1	2n-1
n	n+1	n-1	…	2	2n

Strictly speaking, this ends the procedure - the relevant results table for two dice can be used as a lookup table to determine results. Removing the 1 face from the die on the topmost row of the table - and all the entries in its corresponding column - should be sufficient to produce a set of dice - one of \(n\) sides, one of \(n-1\) sides - that rolls results with the same relative probabilities of two \(n\)-sided dice added together, with the exception of the most likely result which is removed entirely. However, Tatham described a relabeling scheme that makes determing the value of a roll without using a lookup table substantially easier, and for the sake of clarity and completeness it will be presented here.

1. The face on the top die corresponding to the \(n+1\) result should be removed entirely, creating a \(n-1\)-sided die. The column of \(n+1\) results should be removed from the table.
2. The row/column containing the two least likely results - \(2\) and \(2n\) - should be marked with a blank face on the left/top die. On the other die, the row containing a \(2\) result - and the corresponding face on that die - should be marked with \(2\) and one dot. The same should be done for the row containing a \(12\) result.
3. The two second-least likely results - \(3\) and \(2n-1\) - are either in rows or in columns, but oriented the same way as the two least likely results. Mark the row/column containing each result - and the corresponding die faces - with said result, and add two dots to each face.
4. Repeat step 3, this time with the two third least likely results, which should be oriented perpendicular to the results in step 3. Use three dots to mark each face this time, in addition to the result.
5. Repeat steps 3 and 4, alternating between rows and columns, until all faces have been re-marked. Each face will be marked with one more dot than the faces marked on the previous set.

By rolling these two dice and taking the result with the highest number of dots on the face, we can get the same results distribution as found in the table without needing any lookups.