### Idea for a proof of William James Tychonievich's Assumption about Dice

In a previous post, I used a metaphor about rolling all 5 Platonic solid dice, mistakenly believing that the distribution of the sum of the dice rolls would not follow a Bell Curve.  William James Tychonievich mentioned in a comment that he thought it would and has demonstrated this by plotting the data.

Here is an idea for how one might prove why this works.  I suspect that as the number of dice we use increases, the distribution converges to the normal distribution.  However, I have not shown this, but I have shown that the distribution is unimodal (meaning that the probability of the lowest roll is the lowest, then the probability starts to increase until it gets to the highest point, then continues to decrease).  The Bell Curve is a unimodal distribution.

If we roll fair dice, each number on the die has an equal probability.  If we roll multiple dice, then to find the probability of any given roll we multiply the probabilities of each roll.  So, for example, the chance of simultaneously rolling a 1 on a four-sided die (d4), a 3 on a six-sided die (d6), and a 7 on an 8-sided die (d8) is (1/4) x (1/6) x (1/8), which is 1/192.  The reason we multiply the probabilities is because each roll is independent - what we roll on the d4 does not affect what we roll on the d6 or d8.

Likewise, since there are four possible rolls for the first die, 6 for the second, and 8 for the third, there are 192 total possible rolls for the three dice.  Since each roll has an equal probability, we can dispense with the probability and only focus on the rolls themselves.  We can view each die as a list of consecutive integers: (1,2,3,4), (1,2,3,4,5,6), and (1,2,3,4,5,6,7,8).  A roll is then the same as selecting one number from the first list, one from the second, and one from the third.

List each of the numbers in increasing order and consider just the first two lists: (1,2,3,4) and (1,2,3,4,5,6).   Next, generate sums of rolls by starting with 1 and adding 1 to each of the numbers in the second list, then do the same with 2 and so on to generate the following lists:

2, 3, 4, 5, 6, 7

3, 4, 5, 6, 7, 8

4, 5, 6, 7, 8, 9

5, 6, 7, 8, 9, 10

Look at what is happening.  We have 24 rolls and 8 possible sums (from 2 to 10).  Since these are consecutive integers, when we generate the sums in this way, moving fro 1 to 2 only increases by one, so the sums generated by adding 2 to every number of the second list (the first row) are only one greater than those generated by adding one to  the second list (the second row).  So, every number but the first and last overlap.  This pattern continues and at the end,  we see that we get one 2, one 10, two 3's, three 4's, three 8's, and four 5's, 6's and 7's.

For three dice, we do the same thing.  I will not write all 192 rolls, but instead consider the lists: (1,2,3), (4,5,6), (7,8,9).  Generate lists of sums of rolls by first selecting the smallest number from the first list and the smallest from the second and adding these to all the numbers in the third list to form 12, 13, and 14.  Then, add the smallest number in the first list and the second smallest in the second list to form 13, 14, and 15 and continue the process.  Once we exhaust the rolls with 1 from the first list, then use 2, and then 3, following the same process.  Altogether, we generate the following lists:

12, 13, 14

13, 14, 15

14, 15, 16

13, 14, 15

14, 15, 16

15, 16, 17

14, 15, 16

15, 16, 17

16, 17, 18

Once again, when we line up the columns, we see that the sums in the middle occur more often than those on the ends and so this also forms a unimodal distribution.  Observe also that there is nothing special about dice - the important thing is that we are using lists of consecutive integers.  So, in general, if we have m lists of consecutive integers with n1 integers in the first list, n2 in the second, and so on until we have nm in the final list, we can organize the lists in ascending order.  If we generate sums where each sum takes one member from each list in the same way as above, we will create a unimodal distribution.