Find a four-digit number with four different digits, that is equal to the number formed by its digits in descending order minus the number formed by its digits in ascending order.

(In reply to re(2): Interesting! by Gamer)

There's even more to it. I tested numbers with 2, 3, 5 and 6 digits.

1) 2-digits:
We all know that substracting the inverse from a two-digital number gives us a number divisible by 9 and eventually (if repeated or instantly 9) results in 9.

2) Well, the 3-digits are quite boring:
621 - 126 = 495 -> 954
843 - 348 = 495 -> 954
and so on, these are straight numbers, but the sequence eventually ends up with 954 with 495 as "magic number"

3) 5-digits:
any number that produces 98421 and the sequence
98532 (cap the 9 and notice 8532 which was the number prior to the magic 6174)
97443
96642
97731
98532 (from here on it's a loop of 4 numbers)

75321
96642 (this number takes you to the above loop)

54321
97641 (cap 9 and notice the digits in 7641 are the same like the magic digits in the 4-digits-sequence)
98622
97533
96543
97641 (another loop)

8532 and 7641 have significance for the 5-digit-sequence as well, as you can see. Probably there's another loop, I didn't try any further.

4) 6-digits
877542
766431 (results in 631764 which ends the sequence, notice that it once again contains the digits 1,4,6 and 7)
This was fast. Now for a long one:

644331, 885510, 998622, 777321, 655443, 887310, 875322, 765441(hey, there 1,4,6 and 7 again! but this time, it's different...), 876420(watch this number), 875421, 875430, 885420, 886320, 866322, 665442, 876420 (here's the loop)

This time 6174 and it's digits occurs in the number before the one that's actually signifant to produce the loop. Symmetry gone? Coincidence?
That means, we can end up with either the "magic number" or with a loop.

That's it. I guess this works for all infinity. And I assume there are no more sequences which end up only with a magic number as the numbers become longer, but could end up in a loop as well.

Posted by abc on 2003-09-11 18:12:04