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 Reverse and Differ (Posted on 2005-02-01)
Reverse the digits and find the positive difference between the two numbers.
Using this result, continue this process until the difference is a number with fewer than three digits.

1) What are the only possible results? Prove it. 2) What is the greatest number of steps any number could take to terminate (and which numbers are these?)

 Submitted by Jer Rating: 2.0000 (2 votes) Solution: (Hide) 1) 99 is the only posible result. 2) Call the original 3 digits a, b, c. The number itself is 100a + 10b + c and its reversal is 100c + 10b + a The difference is 99a - 99c = 99(a-c) This number must be a multiple of 99, so it is either 99, 198, 297, 396, 495, 594, 693, 792, or 891. 891 - 198 = 693 792 - 297 = 495 693 - 396 = 297 594 - 495 = 99 As can be seen, 99 is always reached. 3) The most steps would result from reaching 891 or 198 on the first step. For example: 422 -> 198 -> 693 -> 297 -> 495 -> 99 which takes 5 steps. Any number whose first and last digits differ by either 2 or 10 will take the full 5 steps. (there are 150 such numbers.)

 Subject Author Date re(4): Solution Jer 2018-03-25 14:01:08 re(3): Solution Ady TZIDON 2013-03-29 19:26:23 re(2): Solution Charlie 2013-03-29 11:42:21 re: Solution Ady TZIDON 2013-03-29 08:52:01 Some thoughts Sing4TheDay 2005-02-02 15:21:53 Solution e.g. 2005-02-01 18:28:11

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