2018
Problem - 4492
Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a < b$ and there are no other
five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b - a$? (A number is a
palindrome if it reads the same forwards and backwards in base $10$.)
Let $a=\overline{xyzyx}$. Then there are three possible cases for $b$ the next palindrome.
- $z < 9$. Then $b=\unerline{xy(z+1)yx}$. Therefore $b-a=100$.
- $z = 9$ and $y < 9$. Then $b=\overline{x(y+1)0(y+1)x}$. Therefore $b-a=110$
- $z=y=9$, Then $b=\overline{(x+1}000(x+1)}$. Therefore $b-a=11$.
So all the possible differences are $$\boxed{11,\ 100,\ 110}$$