Take any four digit number
(whose digits are not all identical), and do the following:
- Rearrange the string of digits to form the
largest and smallest 4-digit numbers possible.
- Take these two numbers and subtract
the smaller number from the larger.
- Use the number you obtain and repeat the above
process.
What happens if you repeat the above process over and over?
Let's see...
Suppose we choose the number 3141.
4311-1134=3177.
7731-1377=6354.
6543-3456=3087.
8730-0378=8352.
8532-2358=6174.
7641-1467=6174...
The process eventually hits 6174 and then stays there!
But the more amazing thing is this: every
four digit number whose digits are not all the same
will eventually hit 6174, in at most 7 steps, and
then stay there!
Presentation Suggestions:
Remember that if you encounter any numbers with fewer than has fewer 4 digits, it must be treated as though it had 4 digits, using leading zeroes. Example: if you start with
3222 and subtract 2333, then the difference is 0999. The next step would then consider the difference 9990-0999=8991, and so on.
You might ask students to investigate what happens for
strings of other lengths or in other bases.
The Math Behind the Fact:
Each number in the sequence uniquely determines the
next number in the sequence. Since there are only
finitely many possibilities, eventually the sequence
must return to a number it hit before, leading to a
cycle. So any starting number will give a sequence
that eventually cycles. There can be many cycles; however,
for length 4 strings in base 10, there
happens to be 1 non-trivial cycle, and it has length 1
(involving the number 6174).
How to Cite this Page:
Su, Francis E., et al. "Kaprekar's Constant."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
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