Searching for Kaprekar's constants: algorithms and results

The number 6174 arises in a semifamous problem in recreational mathematics: take any 4-digit number which uses more than one digit and find the difference between the numbers formed by writing the digits in descending order and ascending order (e.g., starting with 4083 yields 8430− 0348= 8082). Iterate this process using the difference as the new 4-digit number. It was first discovered by the Indianmathematician Kaprekar [5] that this process leads in at most 7 steps to the number 6174, a fixed point of the iteration. 6174 is thus known as Kaprekar’s constant. This curious example reportedly sparked interest on several campuses, including Berkeley and MIT [1]. Several authors have examined the generalized problem using different lengths and/or different bases. In this context, an n-digit number in base b is called a Kaprekar’s constant if it is a fixed point under the descending/ascending difference operation (sometimes called Kaprekar’s routine) and every nontrivial n-digit number eventually is transformed to that fixed point by iterating the operation. Our work continues this line of inquiry, establishing the unique 7-digit (with b = 4) and 9-digit (b = 5) Kaprekar’s constant and shows that there are no 15-, 21-, 27-, or 33-digit Kaprekar’s constants.