Take any four digit number
(whose digits are not all identical), and do the following:
What happens if you repeat the above process over and over?
- 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
Suppose we choose the number 3141.
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!
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.