I don’t post Car Talk’s puzzler often, but I thought this one was fairly good. The answer is on the website and has been read on the air, so no fair looking!
Anyway, here it is.
"Most cars today have odometers that go up to 999,999 miles. In fact, they don’t include tenths of miles anymore. Using an odometer that goes up to 999,999, imagine you drove a car all the way from 0 up to 999,999 miles. In that time, how many times would the number “1” appear? To clarrify, 1,000 and 1,001 would count as two times, with the first one appearing twice.
To repeat, In 999,999, how many times would the number one appear on the odometer?"
1
10
11
21
31
41
51
61
71
81
91
100
Based on this, I will take a WAG at 120,000 times.
I probably wrong, but I have had too much red wine to work it out.
It’d be cheating for me to answer this since I’ve already heard it, but I disagree with the answer provided by Car Talk. I say that the correct number is
[spoiler]63.
Simplified reasoning: it’s the number of times that any dial has actually clicked over to reveal a ‘1’, which is what the wording of question implied was being asked.
Long version: I maintain that the ‘1’ on the dial at 10 miles is no different from the one at 13 miles, or 14, and so on. It IS different from the ‘1’ at 210 miles, though, because by that point it has “disappeared” (having changed to the ‘2’ at 20 miles, the ‘3’ at 30, etc.) and “reappeared” again, and thus counts as a separate appearance. If that’s the case, then the answer is the decimal equivalent of the binary number 111111, which is 32+16+8+4+2+1, or 63.[/spoiler]
I agree with Sublight. Here’s another way of looking at it:
[spoiler]If the odometer has only one digit, ‘1’ will appear once. Shocking but true.
If the odometer has two digits, ‘1’ will appear eleven times (1, 10, 11, 21, 31, 41, 51, 61, 71, 81, 91).
If the odometer has three digits, ‘1’ will appear 111 times: the eleven numbers listed above, prefixed with each of the digits from 0-9 (110 so far), plus an extra one for 100.
If the odometer has four digits, ‘1’ will appear 1111 times: the 111 numbers described above, prefixed with each of the digits from 0-9 (1110 so far), plus an extra one for 1000.
You get the picture. The answer is 111,111.[/spoiler]
Assuming the question means “How many times has the number 1 come up in any position?” Sublight is correct.
The odometer is 1 mile short of one million miles. That means that the units counter has clicked over nearly one million times. Since there are 10 digits in the cycle, the units counter has cycled round 100,000 times - so the number 1 has come up 100,000 times, on the units counter alone.
The 10s counter has cycled round at one-tenth the speed of the units counter, so it has cycled round 10,000 times. That’s another 10,000 appearance of the number 1 in the 10s counter.
Similarly the 100s counter has cycled 1,000 times, the thousands counter 100 times, the 10k counter 10 times and the 100k counter once.
:smack: Gah, you’re right, Santo. I had used binary for my calculation (since ones are the only digits that matter when they appear), but the conversion wasn’t the answer; you have to multiply at each step to account for the digits appearing in all subsequent places. I still have the answer written as “111111” on my written calculations though, and I don’t remember why I did that. More after I work through it again.
ETA: Oh. Because my final answer was NOT in binary, and I’m an idiot. Sublight is correct; Car Talk is not.
Can you show your work? It’d be cool to see how you did it in binary. Is it pretty much identical to Sublight’s method? I’m trying to wrap my head around how to do it in binary, and I can’t really figure it out. Although, I must admit I’m glad you came back to this thread, I’ve been racking my brain over how it could be done for most of the day (off and on, of course. ;)).
ETA: With the method proposed above, there are no duplicates; it only counts a number when it flips.
It’s similar to finding the sum of all numbers from 1 to 100. (1 + 100) + (2 + 99) + … (49 + 52) + (50 + 51). There are 50 groups that add up to 101, ergo, the answer is 5050 (i.e. 50 * 101).
The question as written in the OP is not what they asked on Car Talk. Here’s the actual question:
In the OP he states the following:
In the original question, the thousands digit would be counted individually each time, as 1,000 is a different number from 1,001. Therefore, if you counted the ones in those two numbers the sum would be three.
In the OP, he asserts that the answer is two, as the thousands digit is a repeater.
That is why the answers are different. For the question they asked the Car Talk guys are correct.
