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PUZZLE #7: PALINDROMES IN THE ODOMETER (SOLUTION)

Let the first odometer reading--the one we're trying to determine--be
written *xyabba*. This reflects the assumption that the last four
digits are palindromic, but that *y* does not equal *a*
(which it would do if the
last five digits were palindromic). As we add miles there may be
carry digits, so let's first play it simple and assume *a* is
7 or less. Then, after one mile, the odometer reading will be
*xyabb(a+1)*, whose last five digits are palindromic. This
tells us *y=a+1* and *b=a*; so our old reading must
have been

*x(a+1)aaaa*. Now *a+1* is not 9, so the
odometer reading after two miles must be *x(a+1)aaa(a+2)*.
But now the middle four digits are palindromic, giving us the
contradictory news that *a+1=a*. So our assumption that
*a* is anything but 8 or 9 must have been a wrong one.
So let's take up the case where *a=*8. Then, as above,
since *a* is not 9,
we infer that the original reading was *x98888*. As we
add miles one-by-one up to 2, we get the right palindromes; and
when the third mile is added, we get the reading *x98891*.
Since this is a palindrome, we now have *x=1*, and
198,888 on the odometer
at the outset.

The only issue now is to determine that there are no other solutions
lurking about; i.e., that *a* can't be 9. Indeed, suppose
it were, so that the original reading looked like *xy9bb9*.
Here's where we have to be careful about carry digits. If *b*
were 9 as well, then--because the last 5 digits are not palindromic--
we know that *y* is not 9. Hence one more mile gives the
reading *x(y+1)0000*. But the last five digits now form
a palindrome, making *y+1=0*, an impossibility. So the only
way for *a* to be 9 is for *b* to be no more than 8.
Thus when we add one mile, the reading is *xy9b(b+1)0*,
making *y=0* and *b=8*. This makes the original
reading *x09889*. But then, when we add the second mile,
the reading is *x09891*, with the middle four digits not
forming a palindrome. This last contradiction tells us there are no
solutions other than
198,888.