Palindromic Odometer

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VectorZero
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Location: Kensington

Palindromic Odometer

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

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

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

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?
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.

VectorZero
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Re: Palindromic Odometer

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

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.

VectorZero
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Re: Palindromic Odometer

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

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.

VectorZero
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Re: Palindromic Odometer

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

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

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.

Puck
Posts: 615
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Re: Palindromic Odometer

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.

redrogue
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Re: Palindromic Odometer

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

Spoiler:
Your odometer read 99,999. A klick later, it read 100,000... but the "distance traveled" (1 km) happens to be a palindrome.

uncivlengr
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Re: Palindromic Odometer

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

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

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.

VectorZero
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Re: Palindromic Odometer

Spot on.
Van wrote:Fireballs don't lie.

imatrendytotebag
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Re: Palindromic Odometer

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

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.

uncivlengr
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Re: Palindromic Odometer

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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