Given n dice each with m faces, numbered from 1 to m, find the number of ways to get sum X. X is the summation of values on each face when all the dice are thrown.

The **Naive approach** is to find all the possible combinations of values from n dice and keep on counting the results that sum to X.

This problem can be efficiently solved using **Dynamic Programming (DP)**.

Let the function to find X from n dice is: Sum(m, n, X) The function can be represented as: Sum(m, n, X) = Finding Sum (X - 1) from (n - 1) dice plus 1 from nth dice + Finding Sum (X - 2) from (n - 1) dice plus 2 from nth dice + Finding Sum (X - 3) from (n - 1) dice plus 3 from nth dice ................................................... ................................................... ................................................... + Finding Sum (X - m) from (n - 1) dice plus m from nth dice So we can recursively write Sum(m, n, x) as following Sum(m, n, X) = Sum(m, n - 1, X - 1) + Sum(m, n - 1, X - 2) + .................... + Sum(m, n - 1, X - m)

**Why DP approach?**

The above problem exhibits overlapping subproblems. See the below diagram. Also, see this recursive implementation. Let there be 3 dice, each with 6 faces and we need to find the number of ways to get sum 8:

Sum(6, 3, 8) = Sum(6, 2, 7) + Sum(6, 2, 6) + Sum(6, 2, 5) + Sum(6, 2, 4) + Sum(6, 2, 3) + Sum(6, 2, 2) To evaluate Sum(6, 3, 8), we need to evaluate Sum(6, 2, 7) which can recursively written as following: Sum(6, 2, 7) = Sum(6, 1, 6) + Sum(6, 1, 5) + Sum(6, 1, 4) + Sum(6, 1, 3) + Sum(6, 1, 2) + Sum(6, 1, 1) We also need to evaluate Sum(6, 2, 6) which can recursively written as following: Sum(6, 2, 6) = Sum(6, 1, 5) + Sum(6, 1, 4) + Sum(6, 1, 3) + Sum(6, 1, 2) + Sum(6, 1, 1) .............................................. .............................................. Sum(6, 2, 2) = Sum(6, 1, 1)

Please take a closer look at the above recursion. The sub-problems in RED are solved first time and sub-problems in BLUE are solved again (exhibit overlapping sub-problems). Hence, storing the results of the solved sub-problems saves time.

Following is C++ implementation of Dynamic Programming approach.

// C++ program to find number of ways to get sum 'x' with 'n' // dice where every dice has 'm' faces #include <iostream> #include <string.h> using namespace std; // The main function that returns number of ways to get sum 'x' // with 'n' dice and 'm' with m faces. int findWays(int m, int n, int x) { // Create a table to store results of subproblems. One extra // row and column are used for simpilicity (Number of dice // is directly used as row index and sum is directly used // as column index). The entries in 0th row and 0th column // are never used. int table[n + 1][x + 1]; memset(table, 0, sizeof(table)); // Initialize all entries as 0 // Table entries for only one dice for (int j = 1; j <= m && j <= x; j++) table[1][j] = 1; // Fill rest of the entries in table using recursive relation // i: number of dice, j: sum for (int i = 2; i <= n; i++) for (int j = 1; j <= x; j++) for (int k = 1; k <= m && k < j; k++) table[i][j] += table[i-1][j-k]; /* Uncomment these lines to see content of table for (int i = 0; i <= n; i++) { for (int j = 0; j <= x; j++) cout << table[i][j] << " "; cout << endl; } */ return table[n][x]; } // Driver program to test above functions int main() { cout << findWays(4, 2, 1) << endl; cout << findWays(2, 2, 3) << endl; cout << findWays(6, 3, 8) << endl; cout << findWays(4, 2, 5) << endl; cout << findWays(4, 3, 5) << endl; return 0; }

Output:

0 2 21 4 6

**Time Complexity:** O(m * n * x) where m is number of faces, n is number of dice and x is given sum.

We can add following two conditions at the beginning of findWays() to improve performance of program for extreme cases (x is too high or x is too low)

// When x is so high that sum can not go beyond x even when we // get maximum value in every dice throw. if (m*n <= x) return (m*n == x); // When x is too low if (n >= x) return (n == x);

With above conditions added, time complexity becomes O(1) when x >= m*n or when x <= n.
**Exercise:**

Extend the above algorithm to find the probability to get Sum > X.

This article is compiled by Aashish Barnwal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above