I've been thinking about this for several days, butt still can't prove it though I do believe it.
This is as far as I've got:
In expressions of the form: abcd... a, b, etc are integers in the range 0 to 9 and abcd represents the number with digits a, b, c, d
All numbers are assumed to be in base ten unless otherwise stated.
The transformation T is defined to that T(abcd) = abcd + dcba
The Sum S of any number N is defined as S(N) = sum of the digits of N,
so S(abcd) = a+b+c+d
The 'Hypothesis' is that any positive integer can be converted into a palindrome by a sufficient number of applications of transformation T
I have so far been unable to proved the hypothesis but have several observations that may help in constructing a proof.
(1) The hypothesis has some plausibility, since, if all the digits of some number N are sufficiently small T(N) will be a palindrome.
Suppose N = abcd, then T(N) = abcd + dcba
We evaluate that by performing the additions (d+a), (c+b), (b+c), and (a+d)
If all those sums are less than or equal to 9 they all produce single digits,
so T(abcd) = (a+d)(b+c)(b+c)(a+d) which is a palindrome
For example T(2417) = 2417 + 7142 = 9559
On the other hand, we do not get a palindrome when the addition of two digits gives an answer greater than or equal to 10, because the resulting carrying figure destroys the pattern and more applications of T are then needed to get a palindrome.
(2) It may help to consider the sum of the digits of a number.
If none of the additions involved in calculating T(N) is greater than or equal to 10, then T(N) is a palindrome and
S[T(N)] = 2*S(N)
If v of the additions of digits give an answer greater than or equal to 10, then T(N) is not a palindrome and
S[T(N)] = 2*S(N) - 9v
so there may be a tendency for the digits to get smaller when T(N) is not a palindrome.
(3) The property of palindrome is not an intrinsic property of a number considered in isolation, but only of a number expressed in a particular base.
Every number is a palindrome when expressed in base 1.
In base N, N = 10 and is not a palindrome.