Kaprekar’s constant is a fascinating mathematical concept tied to a specific four-digit number: 6174. Discovered by the Indian mathematician D.R. Kaprekar, it emerges from a simple iterative process applied to almost any four-digit number (with at least two distinct digits). Here’s how it works:
Explanation of Kaprekar’s Constant
- Start with any four-digit number: Choose a number where at least two digits are different (e.g., 1234, not 1111).
- Arrange the digits: Sort them in ascending order (smallest to largest) and descending order (largest to smallest).
- Subtract: Take the larger number (descending order) and subtract the smaller number (ascending order).
- Repeat: Apply the same process to the result, and continue iterating.
Remarkably, after a maximum of 7 iterations, this process almost always leads to 6174, a fixed point where the difference remains 6174. This number is Kaprekar’s constant.
Example
- Start with 3524:
- Descending: 5432
- Ascending: 2345
- Subtract: 5432 – 2345 = 3087
- Next, 3087:
- Descending: 8730
- Ascending: 0378 (or 378)
- Subtract: 8730 – 378 = 8352
- Next, 8352:
- Descending: 8532
- Ascending: 2358
- Subtract: 8532 – 2358 = 6174
- Next, 6174:
- Descending: 7641
- Ascending: 1467
- Subtract: 7641 – 1467 = 6174
At this point, the process stabilizes at 6174, confirming it as a fixed point.
Exceptions
- If all digits are the same (e.g., 1111), the difference becomes 0 immediately, and the process ends there.
- Numbers with fewer than four digits are typically padded with zeros (e.g., 123 becomes 0123), but the core idea applies to four-digit numbers.
Why It’s Interesting
Kaprekar’s constant showcases the beauty of number theory and iterative processes. It’s a “mathematical attractor,” pulling diverse starting points into a single value, which makes it both mysterious and elegant.
This video breaks down the process with examples and explores the quirks of Kaprekar’s routine in an engaging way. Enjoy!