Yes i think there's two components to this and that's the first part. Each digit sum on the left and right side converges to 18. Each side of the equation a -

Yes i think there's two components to this and that's the first part. Each digit sum on the left and right side converges to 18. Each side of the equation a - rev_a = b has digit sums to that iteratively get closer to each other (sometimes stay the same distance but never getting further). Additionally that convergence only happens at 18. Eg. 5200 (sums to 7) - 0025 = 5175 (sums to 18, 11 apart) 7551 (sums to 18) - 1557 = 5994 (sums to 27, 9 apart) 9954 (sums to 27) - 4599 = 5355 (sums to 18, 9 apart) 5553 (sums to 18) - 3555 = 1998 (sums to 27, 9 apart) 9981 (sums to 27) - 1899 = 8082 (sums to 18, 9 apart) 8820 (sums to 18) - 0288 = 8532 (sums to 18, 0 apart) 8532 (sums to 18) - 2358 = 6174 (sums to 18, 0 apart) 7641 (sums to 18) - 1467 = 6174 (sums to 18, 0 apart) I think this is the first clue. The digits can only be equal on each side when they are 18 and the sum of each side progressively gets closer on each side, eventually equalling each other which has to happen at 18. I think if you dive in it's a variation of the classic 'digits sum to 0 mod 9'. Then once the digits on each side sum to 18 i think they must converge onto 6174 from there. So first we have digits always converging to have the same digit sum on each side and that convergence is always when the digit sum is 18 on each side of the equation. I think property is going to be provable by the classic mod 9 rules but it'd take some work. Then i believe we have a second property kicking in that all 4 digit numbers that have digits that sum to 18 on both sides of this equation will converge on 6174. This is a more limited set of numbers. Only numbers of the form a - a_rev = b that have digits that sum to 18 for both a and b need to be considered since we can separately see the convergence to 18 on both sides above.