• Alfred
  • Alfred
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Topic: Kaprekar’s Constant

I’ve just stumbled across a nice little bit of recreational mathematics, so I’ve decided to share it here.

Kaprekar’s Constant is the magical number 6174. It’s named after D.R. Kaprekar, who was an Indian recreational mathematician, and what’s magical about it is that you will always end up with the same number if you follow these steps:

1. Choose any four-digit number in which not all of the digits are the same.
2. Rearrange the digits to form the largest possible number.
3. Rearrange the same digits to form the smallest possible number, including leading zeros if applicable.
4. Subtract the smaller number from the larger one.
5. Repeat the process with the result.

Within seven steps or fewer you will always arrive at 6174, and once you reach that number you’ll be stuck with it (because 7641 - 1467 = 6174).

“Welcome to the present moment. Here. Now. The only moment there ever is.”
― Eckhart Tolle
  • Alfred
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Re: Kaprekar’s Constant

Starting with 1980, the sequence is

9810 - 1089 = 8721
8721 - 1278 = 7443
7443 - 3447 = 3996
9963 - 3699 = 6264
6642 - 2466 = 4176
7641 - 1467 = 6174

“Welcome to the present moment. Here. Now. The only moment there ever is.”
― Eckhart Tolle

3 Reply by GB

  • GB
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Re: Kaprekar’s Constant

very odd!

Credendi non mutant facta
  • Alfred
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Re: Kaprekar’s Constant

Alfred wrote:

Starting with 1980, the sequence is

9810 - 1089 = 8721
8721 - 1278 = 7443
7443 - 3447 = 3996
9963 - 3699 = 6264
6642 - 2466 = 4176
7641 - 1467 = 6174

I forgot to allow leading zeros! The smaller number should always be a straight reversal of the larger number.

Starting with 1980, the correct sequence is much shorter:

9810 - 0189 = 9621
9621 - 1269 = 8352
8532 - 2358 = 6174

“Welcome to the present moment. Here. Now. The only moment there ever is.”
― Eckhart Tolle

5 Reply by GB

  • GB
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Re: Kaprekar’s Constant

As you have shown, making the odd slip doesn't matter. Providing you keep going with the number remaining you will still end up with 6174.

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  • Alfred
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Re: Kaprekar’s Constant

That’s true, Geoff, but the reason behind my decision to start with 1980 was that I wanted to use 1089 because it appears in another series of calculations that involves reversing the digits:

1. Choose any three-digit number in which the first and last digits differ by at least two.
2. Reverse the digits and subtract the smaller number from the larger one.
3. Reverse the digits of the result and add the two numbers.
4. Admire the predictable result!

“Welcome to the present moment. Here. Now. The only moment there ever is.”
― Eckhart Tolle