Calculating the Sum of All Digits from 1 to 2021

The problem of calculating the sum of all digit sums from 1 to 2021 can be approached through a systematic breakdown of the range into subgroups of numbers by their digit length. This article will walk you through the process, detailing how the sum of digits is calculated for single-digit, two-digit, three-digit, and four-digit numbers, and finally, how the total sum is obtained.

1. Understanding the Problem

Our goal is to find the sum of the sums of the digits of every integer from 1 to 2021.

2. Single-Digit Numbers (1-9)

The single-digit numbers are straightforward. The sum of the digits of the numbers from 1 to 9 is simply the sum of the numbers themselves:

1 2 3 4 5 6 7 8 9 45

3. Two-Digit Numbers (10-99)

For two-digit numbers, we can observe that each tens place digit (1-9) appears 10 times, contributing to the sum as follows:

10, 20, 30, ..., 90

Each non-zero units digit (1-9) contributes 9 times, contributing to the sum as follows:

11, 12, 13, ..., 19, 21, 22, ..., 29, ..., 91, 92, ..., 99

The sum of the tens digits is:

10 * (1 2 3 ... 9) 10 * 45 450

The sum of the units digits is:

9 * (1 2 3 ... 9) 9 * 45 405

The total sum of the digits for all two-digit numbers is:

450 405 855

4. Three-Digit Numbers (100-999)

For three-digit numbers, each hundreds digit (1-9) appears 100 times, and each non-zero tens and units digit appears 90 times. Thus, the sum of the digits can be calculated as follows:

100 * (1 2 3 ... 9) 100 * 45 4500

2 * 90 * (1 2 3 ... 9) 2 * 90 * 45 8100

The total sum of the digits for all three-digit numbers is:

4500 8100 12600

5. Four-Digit Numbers in the Range 1000 to 2021

For the range 1000 to 1999, each thousands digit is 1, contributing 1000 times. Each of the other three positions (tens, units, and hundreds digits) has a total sum of the digits as follows:

1000 * 1 100 * (1 2 3 ... 9) 1000 4500 5500

For the range 2000 to 2021, the thousands digit is 2, appearing 22 times:

22 * 2 44

The tens and units positions contribute a total sum of 101 * 245 22 * 1 24522 22 24544

The total sum of the digits for the range 2000 to 2021 is:

44 10 * 90 4 * 22 1 149

6. The Total Sum Calculation

Adding the sums from all the subgroups, we get:

45 (1-9) 855 (10-99) 12600 (100-999) 14500 (1000-1999) 149 (2000-2021) 28149

7. Python Code Implementation

Here is a Python code snippet to calculate the total sum of the digits of all integers from 1 to 2021:

sodall 0 for k in range(1, 2022): sk str(k) sod 0 for n in range(len(sk)): sod int(sk[n]) sodall sod print("The total sum of digits of all integers 1 to 2021 is", sodall)

The output of the code will be:

The total sum of digits of all integers 1 to 2021 is 28149