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The Fantastic 6174

April 1st, 2012 by Math Tricks | 4 Comments | Filed in Mysterious Numbers




The Mystery of 6174

6174. It is a number well known to many. Some say it is mysterious, but I would not go that far. It is, however, a very interesting number. Indeed, the number 6174 is also known as the Kaprekar constant, named after the Indian mathematician Dattaraya Ramchandra Kaprekar who studied the mystery behind 6174.

So what is all the hoopla about 6174? Well, first, if you arrange the digits such that you have the highest number (7641) and also the lowest number (1467), and then determine the difference between the two, you arrive at 6174 (7641 – 1467 = 6174).

Well you say, I suppose this is somewhat interesting. But now suppose you take the number 2355. Do what you did before with 6174 – rearrange the numbers to obtain the highest and lowest numbers, and then subtract the lowest from the highest:

2355 -> 5532, 2355
5532 – 2355 = 3177

Now continue the process with the result:

3177 -> 7731, 1377
7731 – 1377 = 6354

And again:

6354 -> 6543, 3456
6543 – 3456 = 3087

Continuing:

3087 -> 8730, 0378
8730 – 378 = 8352

8352 -> 8532, 2358
8532 – 2358 = 6174

As you can see, this process leads to a convergence to 6174. And as you saw before, performing this routine on 6174 results in an endless loop. Cool!

In fact, this same result will occur for any four digit number you choose as long as the number isn’t composed of the same digit (eg., 4444 would not work). You can even use leading zero’s for your four digit number (eg., 0007). Now that is interesting!

Let’s try another example, one with a lot of zeros – 3000:

3000 -> 3000, 0003
3000 – 3 = 2997

2997 -> 9972, 2799
9972 – 2799 = 7173

7173 -> 7731, 1377
7731 – 1377 = 6354

6354 -> 6543, 3456
6543 – 3456 = 3087

087 -> 8730, 0378
8730 – 378 = 8352

8352 -> 8532, 2358
8532 – 2358 = 6174

It is this process that was discovered by our Indian friend Kaprekar, and is known as Kaprekar’s routine. There are other similar numbers that are obtained for other n-digit numbers. For example, for three digit numbers, this process leads to a convergence to 495. Using the routine on two digit numbers results in an infinite loop:

9 -> 81 -> 63 -> 27 -> 45 -> 9

As an interesting experiment, you are invited to try to determine what happens if you perform this routine on higher digit numbers. You may be surprised to see what your results are!

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