  Consider the following process: Take any four-digit number which has at least
two distinct digits. Then, rearrange the digits of the original number in
ascending and descending order, take these two numbers, and find the difference
between the two. Finally, repeat this routine using the difference as the new
four-digit number. In 1949, D. R. Kaprekar became the first to discover that
this process, known as the Kaprekar Routine, would always yield 6174 within 7
iterations. Since this number remains unchanged after an application of the
Kaprekar Routine, it became known as Kaprekar's Constant. Previous works have
shown that the only base 10 Kaprekar's Constants are 495 and 6174, the 3-digit
and 4-digit case. However, little attention has been given to other bases or
determining which digit cases and which bases have a Kaprekar's Constant. This
paper analyzes the behavior of the Kaprekar Routine in the 3-digit case,
deriving an expression for all 3-digit Kaprekar Constants. In addition, the
author developed a series of C++ programs to analyze the paths integers
followed to their respective Kaprekar's Constant. Surprisingly, it was
determined from this program that the most commonly required number of
iterations required to reach Kaprekar's Constant for 3-digit integers was
consistently 3, regardless of base. When loaded as a matrix, the iteration
requirement data demonstrates a precise recurring relationship reminiscent of
Pascal's Triangle.
