## MATHS BITE: 6174

6174 is known as Kaprekar’s Constant. Why is this number important? Perform the following process (called Kaprekar’s Routine):

1. Take any two digit number whose digits are not all identical.
2. Arrange the digits in descending and then ascending order to get two four digit numbers.
3. Subtract the smaller number from the bigger number.
4. Go to step 2 and repeat.

This process will always reach its fixed point 6174 in at most 7 iterations. 6174 is a fixed point as once it has been reached, the process will continue yielding 7641 – 1467 = 6174.

4311-1134=3177
7731-1377=6354
6543-3456=3087
8730-0378=8352
8532-2358=6174
7641-1467=6174

### The Maths Behind it

Each number in the sequence uniquely determines the next number. As there are only finitely many possibilities, eventually the sequence must return to a number it has already hit. This leads to a cycle.

So any starting number will give a sequence that eventually cycles.

There can be many cycles, but for 4 digit numbers in base 10, there happens to be 1 non – trivial cycle, which involves the number 6174.

## MATHS BITE: Kaprekar Numbers

Consider an n-digit number k. Square it, and then add the right n digits to the left n or n-1 digits (by convention, the second part may start with the digit 0, but must be nonzero). If the result is k then it is called a Kaprekar number. They are named after D. R. Kaprekar, a recreational mathematician from India.

We can extend the definition to any base b:

Let  X  be a non-negative integer and  n a positive integer.  X  is an n-Kaprekar number for base  b  if there exist non-negative integer A, and positive integer B  satisfying:

X2 = Abn + B, where 0 < B < bn
X = A + B

-Wikipedia

### Examples in Base 10

• 297: 2972 = 88209, which can be split into 88 and 209, and 88 + 209 = 297.

• 999: 9992 = 998001, which can be split into 998 and 001, and 998 + 001 = 999.
• In particular, 9, 99, 999… are all Kaprekar numbers.
• More generally, for any base b, there exist infinitely many Kaprekar numbers, including all numbers of the form bn − 1.
• 100: 100 is NOT a Kaprekar number as, although 1002 = 10000 and 100 + 00 = 100, the second part here is zero.

M x

(P.S. Another post on Kaprekar is coming soon!)