Beginning with the decimal representation of any integer N, reverse the digits and add it to N. Iterate this operation. Typically you will soon arrive at a palindrome, i.e., a number that reads the same forwards and backwards.
Some numbers take a long time to yield a palindrome.
In fact, no one knows for sure if ANY number leads to an infinite sequence of palindrome-free numbers in the base 10. However, it isn't hard to prove the existence of sequences that never produce a palindrome in certain other bases. By induction, it follows that the entire sequence consists of repetitions of this cycle, and none of the elements are palindromes. It's interesting that, in each base, all the palindrome-free sequences converge very rapidly on just a small number of sequences. Also, see David Seal's Results .