3-Digit Numbers forming Longest Reverse-and-Add Sequence

Theorem
Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation.

Let $r \left({m}\right)$ be the reverse-and-add process on $m$.

Let $r$ be applied iteratively to $m$.

The $3$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are:
 * $187, 286, 385, 583, 682, 781, 880$

all of which need $23$ iterations.


 * $869$ and $968$ are the result of the first iteration for each of these, and so take $22$ iterations to reach a palindromic number.

Proof
The sequence obtained by iterating $r$ on $187$ is:
 * $187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
 * $17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
 * $47267087164, 93445163438, 176881317877, 955594506548, 170120002107, 8713200023178$

Note that $r \left({187}\right) = r \left({781}\right) = 968$, so the sequence obtained by iterating $r$ on $781$ is the same.

The rest of the numbers converge on the same sequence immediately:


 * $286, 968, \ldots$


 * $385, 968, \ldots$


 * $880, 88, \ldots$


 * $869, 1837, \ldots$

as do their reversals.