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.

- Choose any 4-digit integer whose digits are not all equal: e.g. 4358
- Arrange the digits from in decreasing order: 8543
- Arrange the digits in increasing order: 3458
- Subtract: 8543 – 3458 = 5085
- 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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Guillermo your post reminds me of the Mudd Math Fun Facts site which I think may interest your readers. I have linked to it here:

http://colleenyoung.wordpress.com/2009/08/28/mudd-math-fun-facts/

Thanks Colleen. These are nice tricks. I’ll read them and see if I can make a post about them later. I love the tricks, and I am always interested in discussing the proofs.

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