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Kaprekar's Magic 6174

by olooney on 6/14/24, 1:01 PM with 6 comments

  • by dash2 on 6/14/24, 2:56 PM

    Sort the digits to 'wxyz', where each letter is a 0-9 digit.

    'wxyz' - 'zyxw' = 999(w - z) + 90(x - y)

    w - z is between 1 and 9, since w > z (we have ruled out numbers like 1111). x - y is between 0 and 9. So there are at most 90 such numbers. In fact there are fewer because x-y <= w-z.

    This is why there are many collisions in the first step.

  • by dimastopel on 6/14/24, 3:48 PM

    Numberphile video on the topic: https://youtu.be/d8TRcZklX_Q?si=t9x2HLWYOpPiTbn4

  • by hiperlink on 6/14/24, 2:46 PM

  • by pierrebai on 6/14/24, 8:29 PM

    Interestingly, there is a common pattern for fixed-points and cycles for different number lengths: numbers made of 4, 5 and 9. For example

    length 3: fixed-point 495 length 5: 2-cycle containing 59994 length 6: fixed-point 59994

    Similarly for digits 6, 1, 4, 7:

    length 4: 6174 length 5: 4-cycle containing 61974 length 6: fixed-point 631764

  • by ur-whale on 6/14/24, 2:12 PM

    No attempts at generalization to larger numbers of digits?