Playing Games by Rolling Dice

This is a very short study of how to vary board games by changing the way you use the rolls of two dice. When playing board games with my children, we've found that just adding the dice totals together doesn't always produce the most useful result. Classically (for example in Monopoly), players use the sum of two dice to determine how many spaces to move their counter.  This leads to a very simple distribution of results (die 1 is shown vertically, die 2 is shown horizontally; the sums complete the table). 

Sum	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

Average (mean):  7

Average (mode): 7

Now, for an alternative distribution, we can take the maximum value of the two dice (whichever is the higher of the two).  This only gives a range of 1-6, but it's skewed towards the higher end of the distribution (since there are 11 ways of scoring 6, and only one way of scoring 1).

Max	1	2	3	4	5	6
1	1	2	3	4	5	6
2	2	2	3	4	5	6
3	3	3	3	4	5	6
4	4	4	4	4	5	6
5	5	5	5	5	5	6
6	6	6	6	6	6	6

Mean 4.472222222

Mode 6

Conversely, if we take the minimum of the two values, then we have a distribution which is skewed to the lower end (there are 11 ways of scoring 1, and only one way of scoring 6).

Min	1	2	3	4	5	6
1	1	1	1	1	1	1
2	1	2	2	2	2	2
3	1	2	3	3	3	3
4	1	2	3	4	4	4
5	1	2	3	4	5	5
6	1	2	3	4	5	6

MEAN	2.527778
MODE	1

Comparison

The straight sum gives an average of 7, with a symmetrical split
Taking the maximum (i.e. whichever of the two dice is the largest) gives a mode of 6, and a mean of 4.47.
Taking the minimum gives a mode of 1, and a mean of 2.52.

We found this useful in our games, where we introduce a special feature which allows you to either move your pieces forwards by one roll, or one of your opponents' pieces backwards by one roll.  It's not desirable to move pieces back further than forwards (there's the potential for people to make no forward progress, and extend the game excessively), so the skewed distribution of maximum or minimum is working well for us.

Extension
The distribution of totals for one die gives a mean average of 3.5
For two die, the mean is 7.
For three, I suspect the mean is 10.5, and in a future blog, I'll look at this in more detail, along with other ways of producing interesting distributions with just two dice.