Jun 08, 2009
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.
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.
Answer:
RAY: Here's the answer. Let's look at a single-digit number, you know like one, two, three, four, five. What are the chances that that digit is a one? Well it's one in ten, right, because there are ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
So if there's a one in ten chance that it is a one, there's a nine in ten chance or nine-tenths chance that it isn't. Or point nine (.9).
TOM: Yeah.
RAY: And in fact the chance that all six digits on the odometer reading are not one, is point nine (.9) raised to the power of six. And if you multiply that out, which I did of course,
point nine (.9) to the sixth power comes out point 531441. Or 53 point 1441 percent. OK, so 53.1441% of those million numbers have no one (1).
TOM: Gotcha.
RAY: And if there are a million numbers, 53% of those is 531,441 numbers that contain no one (1) in them.
TOM: Wow.
RAY: Pretty neat, eh? Do we have a winner?
TOM: Yes. We do have a winner. The winner is Don Boshara from Titusville, Florida. And for having his answer selected at random from among all the correct answers that we got, Don is going to get a $26 gift certificate to the Shameless Commerce Division at cartalk.com, with which he can get a brand-new blue on blue Latin T-shirt, that says Non Impediti Ratione Cogitationis.
RAY: Which is the Latin translation of my brother's personal creed and motto: Unencumbered by the Thought Process.
TOM: Absolutely man, absolutely. Congratulations, Don!
So if there's a one in ten chance that it is a one, there's a nine in ten chance or nine-tenths chance that it isn't. Or point nine (.9).
TOM: Yeah.
RAY: And in fact the chance that all six digits on the odometer reading are not one, is point nine (.9) raised to the power of six. And if you multiply that out, which I did of course,
point nine (.9) to the sixth power comes out point 531441. Or 53 point 1441 percent. OK, so 53.1441% of those million numbers have no one (1).
TOM: Gotcha.
RAY: And if there are a million numbers, 53% of those is 531,441 numbers that contain no one (1) in them.
TOM: Wow.
RAY: Pretty neat, eh? Do we have a winner?
TOM: Yes. We do have a winner. The winner is Don Boshara from Titusville, Florida. And for having his answer selected at random from among all the correct answers that we got, Don is going to get a $26 gift certificate to the Shameless Commerce Division at cartalk.com, with which he can get a brand-new blue on blue Latin T-shirt, that says Non Impediti Ratione Cogitationis.
RAY: Which is the Latin translation of my brother's personal creed and motto: Unencumbered by the Thought Process.
TOM: Absolutely man, absolutely. Congratulations, Don!
