The reverse-and-add process is a simple iterative procedure that can be applied to any positive integer. Starting with a number, you reverse its digits and add the reversed number to the original number. This process is repeated with the resulting sum. For example, starting with 59:

  1. Reverse 59 to get 95.
  2. Add 59 and 95 to get 154.
  3. Reverse 154 to get 451.
  4. Add 154 and 451 to get 605.
  5. Reverse 605 to get 506.
  6. Add 605 and 506 to get 1111.

Do we always end up with a palindrome (a number that reads the same forwards and backwards) after a finite number of steps? This is an open question in mathematics. The smallest number for which this question is still unresolved is 196. The trajectory of 196 under the reverse-and-add process has been computed for a billion iterations without ever producing a palindrome, leading to the conjecture that it may never produce one.

The reverse-and-add process can also be applied in binary. Starting with the number 22, which is 10110 in binary, we can compute its trajectory under the binary reverse-and-add process. The sequence of numbers generated by this process exhibits periodic behavior, and it can be shown that it does not produce a palindrome in binary.

Here is a detailed analysis of the binary reverse-and-add trajectory of 22, showing that a palindrome is never produced.