6174 – Kaprekar’s Constant

Here’s a random 4-digit number that I bet you didn’t know was special.

6174. 

Follow these instructions and you will (if you have done things correctly), ALWAYS end up with 6174.

1. Pick any four numbers between 0 – 9. The only rule here is you must pick at least two different numbers. For example, 5566 is fine, but 5556 is not. 
2. Arrange the four numbers in order largest to smallest to create a number.
3. Arrange the four numbers in order smallest to largest to create another number.
4. Take away the smaller number from the larger number.
5. Repeat steps 2 to 4.

Keep doing this and you will notice you will always end up with the same number – 6174.

Let’s do a couple of examples:

Example 1:

Say I chose 8352.

Arrange them largest to smallest – 8532.
Arrange them smallest to largest – 2358.

8532 – 2358 = 6174

If you try to do it for 6174, look what happens:

Arrange them largest to smallest – 7641.
Arrange them smallest to largest – 1467.

7641 – 1467 = 6174

So the process will repeat forever and ever! I’ll save you all a lot of time and just tell you that now!

Example 2:

Say I chose 1989.

Arrange them largest to smallest – 9981.
Arrange them smallest to largest – 1899.

9981 – 1899 = 8082

Arrange them largest to smallest – 8820.
Arrange them smallest to largest – 0288.

8820 – 288 = 8532

Arrange them largest to smallest – 8532.
Arrange them smallest to largest – 2358.

8532 – 2358 = 6174.

And we know what happens next!

The number 6174 has a name. It is known as Kaprekar’s Constant, named after the man who discovered this interesting property. D.R. Kaprekar (full name Dattatreya Ramchandra Kaprekar) was an Indian mathematician, who unfortunately passed away in the mid 1980s. Interestingly, Kaprekar was a school maths teacher, and all of his mathematical discoveries were made in his spare time! 

So there it is, try it for yourself. This will work for any four numbers you choose (where there are at least two different numbers). You will notice that some numbers you choose will take longer to converge to Kaprekar’s Constant. Example 1 from earlier took only two, whereas Example 2 took three. It has been shown that the maximum number of times that it will take is 8. So don’t worry, you won’t be sat there for too long! It’s just down to pot luck how many times you need to repeat the process. 

As an exercise, how many times do you need to repeat the process for the following numbers?

1. 4903
2. 7712
3. 4995
4. 5871
5. 2362

BONUS:

As a bonus, try to work out if the same phenomenon occurs with a 3-digit number. And if so, what is the special 3 digit number that you end up with?

If you think you have it, send it to me via email on sam@metatutor.co.uk and I will let you know if you got it right!

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