# The Kaprekar constant 6174

In the mysterious 495, (1) we chose any 3-digit number, (2) arranged the digits in decreasing order forming the largest integer,  (3) arranged the digit in increasing order forming the smallest integer, and (4) subtracted the smaller from the larger. Each time a difference is obtained, we repeated steps 2-4 several times and we ended up having 495. We explained the mystery behind this ‘phenomenon’ and we were quite fascinated.

In this post, we examine the 4-digit Kaprekar constant. That is, if the digits of a 4-digit number are not all equal, there is a certain number that we will end up with if we repeat the enumerated process above. Let’s have an example.

1. Choose any 4-digit integer whose digits are not all equal:  e.g. 4358
2. Arrange the digits from in decreasing order: 8543
3. Arrange the digits in increasing order: 3458
4. Subtract: 8543 – 3458 = 5085
5. Repeat 2-4 certain number of times.
Repeating 2-4 several times, we get the following computations:
8550 – 0558 = 7992
9972 – 2799 = 7173
7731 – 1377 = 6354
6543 – 3456 = 3087
8730 – 0378  = 8352
8532 – 2358 = 6174
7641 – 1467 = 6174

As we can see, once the number reached 6174, it will keep on repeating.  It is not hard to see that once the digits of the difference contains the four digits 6, 7, 1, and 4 in any order, the next difference will be 6174. Two more examples are shown below.
Example 2: 2011
2110 – 0112 = 1899
9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174

Example 3:  3712
7321 – 1237 = 6084
8640 – 0468 = 8172
8721 – 1278 = 7443
7443 – 3447 = 3996
9963 – 3699 = 6264
6642 – 2466 = 4176
7641 – 1476 = 6174

The question is, like 495, will all the 4-digit number reach 6174 after a certain number of steps? The answer is yes. The proof of this theorem is similar to the proof of the 3-digit kaprekar constant 495.

The number 6174 is called the Kaprekar constant. It was named after Indian mathematicsian Dattaraya Ramchandra Kaprekar.
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