How many numbers between 1 and 1000 begin with the digit 1?

Answer 1

It's not hard to simply count through them manually. The following numbers between one and one thousand begin with the digit 1:

• 1 (one number)
• 10 - 19 (ten numbers)
• 100 - 199 (one hundred numbers)
• 1000 (one number)

That gives us a tally of 1 + 10 + 100 + 1, which equals 112.

More interestingly though, we can see a pattern in how those numbers add up. For example, what if the question was:

"How many numbers between 1 and 1030 begin with the digit 1?"

The answer would be:

111111111111111111111111111112

In other words, for any number that is a power of ten, the quantity of integers between one and that number (inclusive), would be as many digits as the power, and all of those digits would have a value of 1, except the last one, which would be 2.

Or more technically, if x = 10n, then:

      n

d = 1 + Σ 10n

      1

Where d is the number of values from 1 to x that have 1 as the first digit.