## Friday, January 14, 2011

### How Would Someone Figure This Out?

What would possess someone even to think of studying this, much less come up with Kaprekar's Constant?

6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following property:

1. Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)
2. Arrange the digits in ascending and then in descending order to get two four-digit numbers, adding leading zeros if necessary.
3. Subtract the smaller number from the bigger number.
4. Go back to step 2.

The above process, known as Kaprekar's routine, will always reach 6174 in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 3524:

5432 – 2345 = 3087
8730 – 0378 = 8352
8532 – 2358 = 6174

KauaiMark said...

Now that is interesting AND weird.

(I'll be trying it out for a couple of hours just to see if it's a prank or not)

Rhymes With Right said...

Which leads us back to the musical question -- "This is significant because...?"

I mean, other than a barroom trick that will win you a drink or three, what is the significance?

So asks the Social Studies teacher.

KauaiMark said...

"...trick that will win you a drink"

Good enough for me! :)

Mr. W said...

believe it or not when I was in my math classes I was always on my calculator taking numbers flipping them and subtracting them. I don't think I was ever aware of any patterns just screwing around in class I guess.

I remember when I was in the 11th grade and I made an equilateral triangle and then took the midpoints of the sides and connected them. I did it again and was blown away by it. Of course, I later found out I was just copying Sierpinski's triangle.

Joshua Sasmor said...

These constants are attractors in the phase space of arithmetic operations. These are significant in dynamical systems and chaos theory work. One of the most famous (and still unproven) ones is the 3x+1 problem: 1. pick any positive integer. 2. If it is even divide by two, and if it is odd, multiply by three and add 1. 3. Back to step 2. The claim is that this will result in the sequence 4,2,1 for any starting integer. For example, starting with 13 gives: 13, 40, 20 , 10, 5, 16, 8, 4, 2, 1, 4, 2, 1 (ad infinitum). There are LOTS of related problems to this - there's an entire paper about this by J Lagarias (http://www.math.lsa.umich.edu/~lagarias/3x+1.html)