One is the Magic Number

Oct 27, 2008

RAY: This was sent in by Martin Cohen and he says, 'No offense guys but your puzzlers of late have been less than stellar. Not to worry. Help is on the way.'

Odometers on modern cars go from 0 to 999,999. There are no tenths; all the tenths are done by the trip odometer.

Imagine that someone managed to reach 999,999 miles. If you counted all the times that the digit 1 appeared, how many 1s would be counted? In doing this, count each time that 1 appears in a number.

So, for example, you would count 3 times for 1,101. Two times for 1,102. So how many times is the number 1 going to appear in going from 0 to 999,999?
RAY: Here's the answer. How many numbers are there between zero and 999,999?

TOM: Oh a million almost.

RAY: Very good. Each one of those numbers is six digits whether it's 000,001, or 999,973. So if there are a million numbers and each of them is six digits then there must be how many total digits?

TOM: How about six million, eh?

RAY: There you go. And of course there are ten digits between zero and nine, 0, 1, 2, 3, etc. And each of the digits is used as often as each of the others. So if we divide our six million digits by ten we find out that there are 600,000 ones, there are also 600,000 twos, 600,000 threes, because all the digits appear as often as all the other ones.

TOM: Yeah.

RAY: So that's the answer, 600,000. Who's our winner?

TOM: Our winner this week is Meredith Moss from Tempe, Arizona, and for having her answer selected at random from among all the correct answers that we got, Meredith is going to get a $26 gift certificate to the Shameless Commerce Division at, with which she can get a copy of Ask Click and Clack, a book of questions and answers from Car Talk.

RAY: And 600,000 congratulations to you Meredith!

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