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Consider the following process: Take any four-digit number which has at leasttwo distinct digits. Then, rearrange the digits of the original number inascending and descending order, take these two numbers, and find the differencebetween the two. Finally, repeat this routine using the difference as the newfour-digit number. In 1949, D. R. Kaprekar became the first to discover thatthis process, known as the Kaprekar Routine, would always yield 6174 within 7iterations. Since this number remains unchanged after an application of theKaprekar Routine, it became known as Kaprekar's Constant. Previous works haveshown that the only base 10 Kaprekar's Constants are 495 and 6174, the 3-digitand 4-digit case. However, little attention has been given to other bases ordetermining which digit cases and which bases have a Kaprekar's Constant. Thispaper analyzes the behavior of the Kaprekar Routine in the 3-digit case,deriving an expression for all 3-digit Kaprekar Constants. In addition, theauthor developed a series of C++ programs to analyze the paths integersfollowed to their respective Kaprekar's Constant. Surprisingly, it wasdetermined from this program that the most commonly required number ofiterations required to reach Kaprekar's Constant for 3-digit integers wasconsistently 3, regardless of base. When loaded as a matrix, the iterationrequirement data demonstrates a precise recurring relationship reminiscent ofPascal's Triangle.

The Base Dependent Behavior of Kaprekar's Routine: A Theoretical and Computational Study Revealing New Regularities