Not an answer but a very good clue I can see is that the sum of the digits on each side of the equation consistently gets closer and closer. 9541 – 1459 = 8082

Not an answer but a very good clue I can see is that the sum of the digits on each side of the equation consistently gets closer and closer. 9541 – 1459 = 8082 Left hand digit sum = 19. Right = 18. They are 1 apart. 8820 – 0288 = 8532 Both sides now = 18. Now 0 apart and they'll stay there. They are only 0 apart when at 18. 8532 – 2358 = 6174 Both sides = 18 7641 – 1467 = 6174 Both sides = 18 You can play with this a bit and it's consistent. The sum of digits of the left and right hand side consistency get closer to each other iteratively (but not necessarily closer to 18). Eventually they lock in at being equal to each other when their digits sum to 18. This seems to be one property to look at. I think there's then a second thing happening. Once the values on both sides have digits that sum to 18 the process from there converges on to 6174. So first the digits of the two sides to the equation converge to equal the same which always only occurs when the digit sum is 18. The digit sum locks into being at 18 at that point. And then subsequently once the digits are 18 they converge on to 6174. I would start by working out why digits on each side of the equation converge to summing to 18 on both sides of the equation and never being equal at any other value in this process. It reminds me of https://math.stackexchange.com/questions/99725/every-integer-is-congruent-to-the-sum-of-its-digits-mod-9 https://math.stackexchange.com/questions/99725/every-integer... Now the next thing I would do is ask why does every number with digits that sum to 18 eventually end up at 6174. 4 digit numbers with digits that sum to 18 is a very limited set so it should be easy to figure out the combinations and why they all reach 6174. Put those two together and you'd have an answer. (I'm thinking about it now but it really doesn't seem too hard).