Kaprekar's constants — 6174 and 495
Take any four-digit positive whole number. It may start with one or more zeroes but
is must have four digits. The number also must have at least two different digits; i.e., numbers
with four identical digits are not allowed. Next, apply the following steps:
- Sort the digits of the chosen number from left to right in decreasing order. This creates the largest number that
can be made from the digits in the chosen number. Call the result A.
- Sort the digits of the chosen number from right to left in increasing order. This creates the smallest number that
can be made from the digits in the chosen number. Call the result B.
- Subtract B from A. Call the result C.
- Now take C as a newly chosen number and repeat steps 1—4.
At some point —actually after no more than seven iterations— your new C will be 6174. This
is the end of the process because 7641 (new A) - 1467 (new B) = 6174 (new C).
Do the same thing with any number consisting of three digits and the end result is always 495.
6174 and 495 are called Kaprekar constants
after the Indian mathematician
Dattatreya Ramchandra Kaprekar.
The bar graph below shows for all numbers 1 — 9998 (numbers with all identical digits excluded,
of course), how many iterations are required to reach the Kaprekar constant 6174; e.g,
for 2400 of the numbers 1 — 9998 it took three iterations to reach 6174.
Curious to find out what happens with five-digit numbers?
Google it!
Play with four-digit numbers below:
|