## Saturday, January 15, 2011

### 2-in-1: Strange Number 6174 and College Humor for John Ellis

6174 is known as Kaprekar's constant[1][2][3] 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.[4] 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
The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4:
2111 – 1112 = 0999
9990 – 0999 = 8991 (rather than 999 – 999 = 0)
9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174
9831 reaches 6174 after 7 iterations:
9831 – 1389 = 8442
8442 – 2448 = 5994
9954 – 4599 = 5355
5553 – 3555 = 1998
9981 – 1899 = 8082
8820 – 0288 = 8532 (rather than 882 – 288 = 594)
8532 – 2358 = 6174
Note that in each iteration of Kaprekar's routine, the two numbers being subtracted one from the other have the same digit sum and hence the same remainder modulo 9. Therefore the result of each iteration of Kaprekar's routine is a multiple of 9.
495 is the equivalent constant for three-digit numbers. For five-digit numbers and above, there is no single equivalent constant; for each digit length the routine may terminate at one of several fixed values or may enter one of several loops instead.[4]

## References

1. ^ Mysterious number 6174
2. ^ Kaprekar DR (1955). "An Interesting Property of the Number 6174". Scripta Mathematica 15: 244–245.
3. ^ Kaprekar DR (1980). "On Kaprekar Numbers". Journal of Recreational Mathematics 13 (2): 81–82.
4. ^ a b

#### 3 comments:

Steven Colyer said...

Dangit, Heidi, stop invading my posts! How do you DO that anyway? Is your middle name Feynman?

Pat B said...

Steven,
I can remember spending hours tracing the "orbits" of numbers after reading about Kaprekar's number. I would make up rules for transforming numbers and then see how they are linked.... and that is no penguin, that is a milkshake with two straws...

Steven Colyer said...

Really? I never heard of that number or that problem. Yahoo or CNN, I forget which has a list of the ten strangest things in Wikipedia, that's how I found out about it. And oh yeah that's a milkshake, hmm.

I thought the list of constants at the bottom was cool, I got lost there for an hour or so.