Ken's POTW 020103

Series of Reversed Differences

  abc
- cba
-----
  xyz

Beginning with a three-digit number, reverse the order of the digits and subtract the smaller number from the larger. If the result is a three-digit number, repeat until the result has less than three digits or is a duplicate of a previous result.

What is the longest series of subtractions before reaching a two-digit number or a duplicate?
If a cycle exists (a duplicate result is found), what is the longest cycle?
What is the most likely 2-digit result to end the series? If we continue to reverse and subtract, what is the most common 1-digit result?

Repeat the problem for 2-digit numbers and for 4-digit numbers.

Source: Original.

Solutions were received from Al Zimmermann, Denis Borris, David Peck, Joseph DeVincentis, Dane Brooke, Samantha Levin, David Arnold, Ross Millikan, and Sudipta Das. Samantha Levin's submission is a good summary, with some examples from Ross Millikan and Joseph DeVincentis:

Very interesting puzzle! In the 2 (3) digit cases, everything degenerates to 9 (99) - there are no cycles. The longest series is 6 (5 not counting the final 1 or 2 digit number) in both cases (achieved by 16 different 2-digit numbers and 160 different 3-digit numbers).

[One example of each from Ross Millikan: 654->198->693->297->495->99. 35->18->63->27->45->9]

In the 4-digit case, there is one cycle of length 2: 2178, 6534. Some numbers degenerate to this cycle; all other numbers degenerate to 999, with the longest series being 13 (achieved by 173 different numbers).

[One example from Joseph DeVincentis: This would be satisfied by a number such as 3800, which follows the path 3717, 3456, 3087, 4716, 1458, 7083, 3276, 3447, 3996, 2997, 4995, 999.]

For higher numbers of digits, there are also cycles of length 2, which are variations of the 2178, 6534 cycle:
21(any number of 9's)78, 65(same number of 9's)34 is a 2-cycle
21782178,65346534 is a 2-cycle (etc. for 3,4,.... repeats)

The only other cycle I found was a cycle of length 14 for 8-digit numbers, generated by 11436678.

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