The sum of all digits from 1 to a million

Question

Every whole number from 1 to 1,000,000 is written out, and all the individual digits are added together. What is the total? Find a method that avoids any long addition, and be careful with the endpoint.

Accepted Answer

27,000,001. The six-digit strings. Consider all strings from 000000 to 999999: one million strings, six positions each. In any single position, each digit 0 to 9 appears exactly 100,000 times (once for each of the 100,000 choices of the other five positions). The digits 0 to 9 sum to 45, so one position contributes 45 × 100,000 = 4,500,000, and six positions contribute 6 × 4,500,000 = 27,000,000. Padding zeros add nothing, so this is the digit sum of 1 to 999,999. The endpoint. The number 1,000,000 has digit sum 1. Total: 27,000,001. A pairing check. Pair each n from 0 to 999,999 with 999,999…

The sum of all digits from 1 to a million

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