6174

6174, also known as Kaprekar's constant, is a fixed point of a digit-rearrangement routine applied to four-digit numbers in base 10. It arises when, starting from any four-digit number that does not have all identical digits, one repeatedly forms two numbers from its digits—one in descending order and one in ascending order—subtracts the smaller from the larger, and repeats the process with the result.

The procedure is as follows: take a four-digit number with at least two distinct digits, arrange its digits to create the largest and smallest possible four-digit numbers (leading zeros are allowed in the ascending form), subtract the smaller from the larger, and repeat with the result. After several iterations, the result converges to 6174, and applying the same steps again yields 6174, entering a fixed point. If the starting number has all identical digits, the process yields 0000 and remains there, which is a separate degenerate case.

6174 was discovered by Indian mathematician D. R. Kaprekar in 1949 while studying these routines. It is the unique fixed point for the base-10, four-digit Kaprekar operation. In practice, most starting numbers reach 6174 within at most seven iterations. The constant has become a well-known example in recreational mathematics and has motivated exploration of similar routines in other digit lengths and numeral bases, such as 495 for three-digit numbers.