Team selection tests for JBMO 2018

Question

Let M and N are two nine-digit palindromes without leading zeros (a palindrome is a number that reads the same backward as forward) such that M < N and there are no other nine-digit palindromes strictly between M and N. What are all possible values of N - M?

Accepted Answer

Denote the number M is a b c d e d c b a. We consider cases: 1. If e 8 then the next number N is a b c d (e+1) d c b a. 2. If e = 9, d 8 then the next number N is a b c (d+1) 9 (d+1) c b a. 3. If e = d = 9, c 8 then the next number N is a b (c+1) 999 (c+1) b a. 4. If e = d = c = 9, b 8 then next number is a (b+1) 99999 (b+1) a. 5. If e = d = c = b = 9, then a 8 (otherwise, there is no N). Then the next number is (a+1) 9999999 (a+1). So the difference are 10000, 10100, 1000100, 10000010, 100000001.