## If the numbers $1$ to $5$ are written out in words: one, two, three, four, five, then there are $3 + 3 + 5 + 4 + 4 = 19$ letters used in total.

If all the numbers from $1$ to $1000$ (one thousand) inclusive were written out in words, how many letters would be used?

NOTE: Do not count spaces or hyphens. For example, $342$ (three hundred and forty-two) contains $23$ letters and $115$ (one hundred and fifteen) contains $20$ letters. The use of “and” when writing out numbers is in compliance with British usage.

### To compute this exactly would be a lengthy effort, but we can make the computation easier by breaking the problem down into smaller parts.

Let’s first focus on one to nine.

1: “one” has 3 letters.

2: “two” has 3 letters.

3: “three” has 5 letters.

4: “four” has 4 letters.

5: “five” has 4 letters.

6: “six” has 3 letters.

7: “seven” has 5 letters.

8: “eight” has 5 letters.

9: “nine” has 4 letters.

If we sum these up, we get 3 + 3 + 5 + 4 + 4 + 3 + 5 + 5 + 4 = 36 letters.

Now let’s look at ten to nineteen.

10: “ten” has 3 letters.

11: “eleven” has 6 letters.

12: “twelve” has 6 letters.

13: “thirteen” has 8 letters.

14: “fourteen” has 8 letters.

15: “fifteen” has 7 letters.

16: “sixteen” has 7 letters.

17: “seventeen” has 9 letters.

18: “eighteen” has 8 letters.

19: “nineteen” has 8 letters.

If we sum these up, we get 3 + 6 + 6 + 8 + 8 + 7 + 7 + 9 + 8 + 8 = 70 letters.

Let’s look at twenty to ninety-nine. We have ten occurrences of twenty, thirty, forty, etc. up to ninety, and then each group has the numbers one to nine appended to it. Twenty to ninety when written gives 6, 6, 5, 5, 5, 5, 5, 7 respectively.

These total up to 44 letters, but we have these ten times, giving 440 letters. We also have 80 instances of the numbers one to nine (since we have eight groups; twenty, thirty, and so on up to ninety), and each group contains the numbers one to nine 10 times. We’ve already calculated that one to nine sums to 36 letters. Therefore, 36 * 80 = 2880 letters.

Adding nine hundred ninety-nine instances of “and”, which adds 3 letters per instance, would give 2997 letters.

When we have “hundred”, note that we have 100 instances of “one hundred”, “two hundred”,…, “nine hundred”. Each “hundred” adds 7 letters, so we have 700 letters. But we’re going to repeat these for each occurrence of one to nine, so in total we have 700 * 10 = 7000 letters.

Adding one to ninety-nine up (since these occur ten times) gives us:

(36 + 70 + 440 + 2880) * 10 = 34260 letters.

Finally, we have “one thousand”, which uses 11 letters.

Adding these all up gives: 34260 letters (for numbers 1 to 99, repeated 10 times) + 2997 letters (for all the “and”s) + 7000 letters (for the “hundred”s) + 11 letters (“one thousand”) = 42268 letters.

So, if all the numbers from 1 to 1000 inclusive were written out in words, 42268 letters would be used.

##### More Answers:

Longest Collatz SequenceLattice Paths

Power Digit Sum