# Number One

RAY: This is from my higher mathematics series and it's a twist on a puzzler I gave some months ago. The idea for this puzzler was sent in by Tim Davis.

He writes:

"How many times does the mileage on an odometer not contain the number 1 at all?"

For example 999,999 doesn't have it. So the question is, how many times does the mileage appear going from 000000 to all 9's (999,999), with no 1s at all?

To refresh your memories, we had a puzzler last Fall asking how many times the number 1 will appear on the odometer that goes from all zeroes, 000000, to all nines, 999999, once it completely turns over. For example at mile 000111, the number 1 appears three times.

The answer to that puzzler is 600,000 times. There are a million numbers to get from all zeroes to all nines. And each one of those numbers is six digits. For example the number 100,000 is six digits, one 1 and five 0s. And because there are a million numbers and each of them is six digits, there are six million digits used.

The number 1 appears 1/10th of those times, because there are ten digits, 0, 1 , 2, 3, 4, 5, 6, 7, 8, 9, and there must be 600,000 ones that appear, in those million numbers, as well as there are 600,000 twos and threes and so forth.

Think you know? Drop Ray a note!

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