Palindromic Odometer

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VectorZero
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Palindromic Odometer

Postby VectorZero » Thu May 06, 2010 3:36 pm UTC

My car's 20 years old, but it's been looked after fairly well (single owner from new, sunday drives to church ... at least until I got my hands on it two years ago) Anyway, I was driving home from work recently, checked the odometer (nearly due for a service) and noticed that the distance travelled was a palindrome.

Kinda neat.

Anyway, I continued on for another kilometre, only to find the odometer was showing a different palindrome.

How far had my car travelled?

Spoiler:
And no silly buggers claiming miles vs kilometres; all units are metric.


Spoiler:
Another neat odometer reading: a palindromic number, itself comprised of concatenated palindromes, both of which had 2 axes of reflective and 2-fold rotational symmetry
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JBJ
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Re: Palindromic Odometer

Postby JBJ » Thu May 06, 2010 3:44 pm UTC

Spoiler:
Simplest explanation is that it rolled over. i.e. 999999 rolled over to 000000
Are you also counting 1/10th's of a kilometer? Or just the whole number?
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VectorZero
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Re: Palindromic Odometer

Postby VectorZero » Thu May 06, 2010 3:57 pm UTC

JBJ wrote:
Spoiler:
Simplest explanation is that it rolled over. i.e. 999999 rolled over to 000000
Are you also counting 1/10th's of a kilometer? Or just the whole number?
Spoiler:
That is a valid answer, and I intentionally phrased the question in a way as to not exclude it for reasons of 169. Those were, however, not the numbers I saw, both of which where whole numbers (my odometer displays only whole numbers, the trip meter displays tenths.) The preamble may give you a clue, and the spoiler will definitely give a ballpark number.
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Puck
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Re: Palindromic Odometer

Postby Puck » Thu May 06, 2010 4:07 pm UTC

Hmm... I'm not sure I understand it, then.

You traveled one kilometer, so the ones digit of the number increments.

For the number to remain a palindrome, the most significant digit must then also increment.

The only way this can occur is if the ones digit is incrementing from 9 to 0 (and therefore so is every other digit).

QED?
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VectorZero
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Re: Palindromic Odometer

Postby VectorZero » Thu May 06, 2010 4:09 pm UTC

This actually happened: the preamble is all true. It's a little 169, however.
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thomblake
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Re: Palindromic Odometer

Postby thomblake » Thu May 06, 2010 4:15 pm UTC

Spoiler:
The "reading" is some number of digits including leading 0's. If the reading is a palindrome, then the first and last digit are the same. Unless the number is all 9's, changing the last digit will not change the first digit. Therefore, if x is a palindrome, then x+1 is not a palindrome, or x is all 9's.

If I'm wrong, then it's in the definition of "reading" or "palindrome".

If you don't count leading 0's, there are trivial cases from x=0 to 8.

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VectorZero
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Re: Palindromic Odometer

Postby VectorZero » Thu May 06, 2010 4:19 pm UTC

Spoiler:
Well, yes, but there is also (i think) 1 non-trivial case.
Ok, maybe it's VERY 169.
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thomblake
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Re: Palindromic Odometer

Postby thomblake » Thu May 06, 2010 4:21 pm UTC

Well, a rather 169 answer:
Spoiler:
After checking the odometer, you noticed the "distance traveled" is a palindrome. If the odometer rolled over in the past and goes up to 99999, then you might have read "11111" on the odometer and known it was really "111111" in "distance traveled". Then you checked the odometer again and noticed the reading was itself a palindrome.

And then the solution would be whatever satisfies both of those. Say, "1221" and "222" for a 3-digit odometer.

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VectorZero
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Re: Palindromic Odometer

Postby VectorZero » Thu May 06, 2010 4:37 pm UTC

thomblake wrote:Well, a rather 169 answer:
Spoiler:
After checking the odometer, you noticed the "distance traveled" is a palindrome. If the odometer rolled over in the past and goes up to 99999, then you might have read "11111" on the odometer and known it was really "111111" in "distance traveled". Then you checked the odometer again and noticed the reading was itself a palindrome.

And then the solution would be whatever satisfies both of those. Say, "1221" and "222" for a 3-digit odometer.
Spoiler:
Nice one, I hadn't thought of the effect of rolling over. But this applies to the situation described; you can assume a 6-digit odometer with a car which has seen only regular use and has not rolled over.
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Aardvarki
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Re: Palindromic Odometer

Postby Aardvarki » Thu May 06, 2010 5:01 pm UTC

Well, if you're counting the tenths it could be something like this:
Spoiler:
29,999.2

Drive for a kilometer (or maybe a tad more), look down:

30,000.3
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rigwarl
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Re: Palindromic Odometer

Postby rigwarl » Thu May 06, 2010 5:02 pm UTC

Spoiler:
Perhaps your odometer read "011110" which is a palindrome, then went up to "011111", where the distance traveled is a palindrome?

It's pretty much the same as thomblake's answer, but doesn't involve rolling over.

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Re: Palindromic Odometer

Postby Puck » Thu May 06, 2010 5:24 pm UTC

Wait a minute...

Spoiler:
Do all the digits of the odometer work correctly?

If some of the number wheels aren't working right, there are potentially a number of ways this could be true.
22/7 wrote:If I could have an alternate horn that would yell "If you use your turn signal, I'll let you in" loud enough to hear inside another car, I would pay nearly any amount of money for it.

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Re: Palindromic Odometer

Postby redrogue » Thu May 06, 2010 8:27 pm UTC

Honest answer:
Spoiler:
I'm guessing it read 99,999, then 100,001. Rounding error (99,999.49 vs. 100,000.50), I'm suspecting.


169 Answer:
Spoiler:
Your odometer read 99,999. A klick later, it read 100,000... but the "distance traveled" (1 km) happens to be a palindrome.
Is 'no' your answer to this question?

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Re: Palindromic Odometer

Postby uncivlengr » Thu May 06, 2010 8:39 pm UTC

redrogue wrote:
Spoiler:
I'm guessing it read 99,999, then 100,001. Rounding error (99,999.49 vs. 100,000.50), I'm suspecting.
Spoiler:
to go from 99999 to 100001, you would have needed to travel more than 1 km. Odometers don't round to the nearest whole number, so if you've travelled 34.9 km, the odometer reads 34.


To the OP:
Spoiler:
By "travelled one km", do you mean the odometer showed a difference of 1 km between the first and second palindrome?
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VectorZero
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Re: Palindromic Odometer

Postby VectorZero » Fri May 07, 2010 12:21 am UTC

rigwarl wrote:
Spoiler:
Perhaps your odometer read "011110" which is a palindrome, then went up to "011111", where the distance traveled is a palindrome?

It's pretty much the same as thomblake's answer, but doesn't involve rolling over.
Spoiler:
This is the closest to my answer so far; however, note the odometer displays a palindrome at the end of the vignette, not at the beginning.
uncivlengr wrote:To the OP:
Spoiler:
By "travelled one km", do you mean the odometer showed a difference of 1 km between the first and second palindrome?
Spoiler:
Yes. No funny business, normal working odometer (which clicks over in about 50m at the end of each kilometre,) and the distance travelled was as close to 1km as makes no difference
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lordatog
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Re: Palindromic Odometer

Postby lordatog » Fri May 07, 2010 4:15 am UTC

Spoiler:
At first, it reads 098889, and the distance travelled, 98889, is a palindrome. After 1 km, it reads 098890, and the odometer reading is now a palindrome.

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VectorZero
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Re: Palindromic Odometer

Postby VectorZero » Fri May 07, 2010 5:29 am UTC

Spot on.
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imatrendytotebag
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Re: Palindromic Odometer

Postby imatrendytotebag » Fri May 07, 2010 7:53 am UTC

Spoiler:
I wouldn't call that 169. Tricky wording, maybe. But the answer is definitely gettable, and makes sense.
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levantis
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Re: Palindromic Odometer

Postby levantis » Tue May 25, 2010 8:32 pm UTC

If you had travelled almost the same ditance backwards, your odometer might read a number between 0 and 8. being inreased by 1, it`s still a palindrome.

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Re: Palindromic Odometer

Postby uncivlengr » Wed May 26, 2010 3:43 pm UTC

levantis wrote:If you had travelled almost the same ditance backwards, your odometer might read a number between 0 and 8. being inreased by 1, it`s still a palindrome.

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