Top Dice

There are $1111$ ways in which five $6$-sided dice (sides numbered $1$ to $6$) can be rolled so that the top three sum to $15$. Some examples are:

$D_1,D_2,D_3,D_4,D_5 = 4,3,6,3,5$

$D_1,D_2,D_3,D_4,D_5 = 4,3,3,5,6$

$D_1,D_2,D_3,D_4,D_5 = 3,3,3,6,6$

$D_1,D_2,D_3,D_4,D_5 = 6,6,3,3,3$

In how many ways can twenty $12$-sided dice (sides numbered $1$ to $12$) be rolled so that the top ten sum to $70$?

The problem given seems complex since it involves 20 dice, but it can be solved by using dynamic programming and combinatorics.

Let’s solve a similar simpler case: How many ways can three 6-sided dice be rolled so the sum of them is 10?

First, create a table to track all the possibilities. In this case, the table will have 4 rows (representing the three dice plus one extra for convenience). The columns will range from 0 to 10 (the required sum). In the first row, fill with ones since there is one way to achieve any sum with zero dice (which is, by rolling no dice).

The subsequent entries in the table represent how many ways we can reach a particular sum by rolling that many dice. For example, there are 3 ways to roll a 5 with two dice (2+3, 3+2, 4+1), so entry (2,5) in the table will be 3.

The final answer will be in the bottom right of the table, which gives the total count for three dice to sum up to 10.

Take this approach to solve the original problem:

There are 20 dice, each with 12 faces, so create a 2D table with 21 rows and 71 columns. The first row should start with “1” and be followed by zeros. Subsequent entries will represent how many ways we can reach a particular sum with a particular number of dice.

Start filling the table iteratively, with the entry at row i and column j being the total of:

– The entry to its left (immediately above in row i-1), which represents the number of ways to achieve the same sum without the current die, and,
– The entry h steps above, where h is the face value of the die, which represents the number of ways to achieve the sum by including at least one die with that face value.

By the end, the value at the bottom right (row 21, column 70) represents the total number of ways the top ten dice can sum to 70.

Since the problem asks for the top ten dice, consider only the partial sums from 1 to 10 dices and once you get the total number of ways for every possible sum for each amount of dices then pick only the subsets that add up to 70.

Note: This algorithm assumes that the order of the dice doesn’t matter. If the order of the dice does matter, then you will need to multiply the final answer by the number of ways to arrange 20 dice with ten of them being part of the sum.
Unfortunately, due to its size, this calculation is better performed with a computer program.

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