Consider repeatedly adding a reverse (of digits) of a number to itself, until a palindrome is reached. Numbers that never reach a palindrome (diverging to infinity), are called Lychrel numbers.
Let's modify this procedure such that after the addition, the first digit is removed if it is equal to $1$. Notice that the modified procedure can't increase the number of digits. That is, we prevented the procedure from diverging. Now, every number either reaches a palindrome or gets stuck in a cycle.
For base $10$ (decimal), the smallest number that gets stuck in a cycle is $10039547$, cycling over a set of $8$ numbers, starting with $84632548$. This was discussed in Conjecture related to Lychrel numbers. For base $2$ (binary), cycles are not possible. That is, in binary, a number can't maintain its number of digits for more than two iterations.
Now the question is, which number bases $b\gt 2$ have cycles?
I have found cycles and proven families of cycles for a lot of number bases $b$. (For example, number bases of form $b=31k-9, b=31k-6, k\in\mathbb N$ will have at least $2$ cycles cycling over $5$ four digit numbers.)
Motivated by small examples, such as the mentioned base $10$ example:
$$ 84632548 = (8, 4, 6, 3, 2, 5, 4, 8)_{10} = (x, y, z, w, w-1, z-1, y, x)_b, $$
I have searched for cycles containing a number represented by one of the following forms:
$$ \begin{array}{} (x, & y, & y-\delta_y, & x-\delta_x)_b \\ (x, & y, & z, & y-\delta_y, & x-\delta_x)_b \\ (x, & y, & z, & z-\delta_z, & y-\delta_y, & x-\delta_x)_b \\ (x, & y, & z, & w, & z-\delta_z, & y-\delta_y, & x-\delta_x )_b \\ (x, & y, & z, & w, & w-\delta_w, & z-\delta_z, & y-\delta_y, & x-\delta_x )_b \\ \dots \end{array} $$
where $x,y,z,\dots\lt b, x\gt 0$ are first half of digits and $\delta_x,\delta_y,\delta_z,\dots\in\{-1,0,1\}$ are offsets in second half of digits. That is, "almost palindromes" are good candidates for cycles.
For example, using this, I've found cycles for all bases $b\le 30$ except for $b\in\{3, 4, 29\}$.
It seems all $b\gt 2$ will eventually have cycles. Any hope in proving this?