Add up every digit from 1 to 999

Question

Write out all the whole numbers from 1 to 999 and add up every single digit that appears. (So 999 contributes 9 + 9 + 9 = 27, and 10 contributes 1 + 0 = 1.) What is the grand total? Find a way to do it without adding 999 numbers.

Accepted Answer

13,500. Pad to three digits. Adding leading zeros changes nothing, since a zero contributes 0, and including 000 changes nothing either. So consider all 1,000 three-digit strings from 000 to 999. One position at a time. Look at the units position across all 1,000 strings. Each digit 0 to 9 appears in that position exactly 100 times (once for each of the 100 choices of the other two digits). The digits 0 to 9 add to 45, so the units position contributes 45 × 100 = 4,500. The tens position and the hundreds position each contribute the same, by the same argument. Total: 3 × 4,500 = 13,500. A pai…

Add up every digit from 1 to 999

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