Given a number x, find sum of digits in all numbers from 1 to n.

Examples:

Input: n = 5 Output: Sum of digits in numbers from 1 to 5 = 15 Input: n = 12 Output: Sum of digits in numbers from 1 to 12 = 51 Input: n = 328 Output: Sum of digits in numbers from 1 to 328 = 3241

**Naive Solution:**

A naive solution is to go through every number x from 1 to n, and compute sum in x by traversing all digits of x. Below is C++ implementation of this idea.

// A Simple C++ program to compute sum of digits in numbers from 1 to n #include<iostream> using namespace std; int sumOfDigits(int ); // Returns sum of all digits in numbers from 1 to n int sumOfDigitsFrom1ToN(int n) { int result = 0; // initialize result // One by one compute sum of digits in every number from // 1 to n for (int x=1; x<=n; x++) result += sumOfDigits(x); return result; } // A utility function to compute sum of digits in a // given number x int sumOfDigits(int x) { int sum = 0; while (x != 0) { sum += x %10; x = x /10; } return sum; } // Driver Program int main() { int n = 328; cout << "Sum of digits in numbers from 1 to " << n << " is " << sumOfDigitsFrom1ToN(n); return 0; }

Output

Sum of digits in numbers from 1 to 328 is 3241

**Efficient Solution:**

Above is a naive solution. We can do it more efficiently by finding a pattern.

Let us take few examples.

sum(9) = 1 + 2 + 3 + 4 ........... + 9 = 9*10/2 = 45 sum(99) = 45 + (10 + 45) + (20 + 45) + ..... (90 + 45) = 45*10 + (10 + 20 + 30 ... 90) = 45*10 + 10(1 + 2 + ... 9) = 45*10 + 45*10 = sum(9)*10 + 45*10 sum(999) = sum(99)*10 + 45*100

In general, we can compute sum(10^{d} – 1) using below formula

sum(10^{d}- 1) = sum(10^{d-1}- 1) * 10 + 45*(10^{d-1})

In below implementation, the above formula is implemented using dynamic programming as there are overlapping subproblems.

The above formula is one core step of the idea. Below is complete algorithm

**Algorithm: sum(n)**

1) Find number of digits minus one in n. Let this value be 'd'. For 328, d is 2. 2) Compute some of digits in numbers from 1 to 10^{d}- 1. Let this sum be w. For 328, we compute sum of digits from 1 to 99 using above formula. 3) Find Most significant digit (msd) in n. For 328, msd is 3. 4) Overall sum is sum of following terms a) Sum of digits in 1 to "msd * 10^{d}- 1". For 328, sum of digits in numbers from 1 to 299. For 328, we compute 3*sum(99) + (1 + 2)*100. Note that sum of sum(299) is sum(99) + sum of digits from 100 to 199 + sum of digits from 200 to 299. Sum of 100 to 199 is sum(99) + 1*100 and sum of 299 is sum(99) + 2*100. In general, this sum can be computed as w*msd + (msd*(msd-1)/2)*10^{d}b) Sum of digits in msd * 10^{d}to n. For 328, sum of digits in 300 to 328. For 328, this sum is computed as 3*29 + recursive call "sum(28)" In general, this sum can be computed as msd * (n % (msd*10^{d}) + 1) + sum(n % (10^{d}))

Below is C++ implementation of above aglorithm.

// C++ program to compute sum of digits in numbers from 1 to n #include<bits/stdc++.h> using namespace std; // Function to computer sum of digits in numbers from 1 to n // Comments use example of 328 to explain the code int sumOfDigitsFrom1ToN(int n) { // base case: if n<10 return sum of // first n natural numbers if (n<10) return n*(n+1)/2; // d = number of digits minus one in n. For 328, d is 2 int d = log10(n); // computing sum of digits from 1 to 10^d-1, // d=1 a[0]=0; // d=2 a[1]=sum of digit from 1 to 9 = 45 // d=3 a[2]=sum of digit from 1 to 99 = a[1]*10 + 45*10^1 = 900 // d=4 a[3]=sum of digit from 1 to 999 = a[2]*10 + 45*10^2 = 13500 int *a = new int[d+1]; a[0] = 0, a[1] = 45; for (int i=2; i<=d; i++) a[i] = a[i-1]*10 + 45*ceil(pow(10,i-1)); // computing 10^d int p = ceil(pow(10, d)); // Most significant digit (msd) of n, // For 328, msd is 3 which can be obtained using 328/100 int msd = n/p; // EXPLANATION FOR FIRST and SECOND TERMS IN BELOW LINE OF CODE // First two terms compute sum of digits from 1 to 299 // (sum of digits in range 1-99 stored in a[d]) + // (sum of digits in range 100-199, can be calculated as 1*100 + a[d] // (sum of digits in range 200-299, can be calculated as 2*100 + a[d] // The above sum can be written as 3*a[d] + (1+2)*100 // EXPLANATION FOR THIRD AND FOURTH TERMS IN BELOW LINE OF CODE // The last two terms compute sum of digits in number from 300 to 328 // The third term adds 3*29 to sum as digit 3 occurs in all numbers // from 300 to 328 // The fourth term recursively calls for 28 return msd*a[d] + (msd*(msd-1)/2)*p + msd*(1+n%p) + sumOfDigitsFrom1ToN(n%p); } // Driver Program int main() { int n = 328; cout << "Sum of digits in numbers from 1 to " << n << " is " << sumOfDigitsFrom1ToN(n); return 0; }

Output

Sum of digits in numbers from 1 to 328 is 3241

The efficient algorithm has one more advantage that we need to compute the array ‘a[]’ only once even when we are given multiple inputs.

This article is computed by **Shubham Gupta**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above