6174 is known as Kaprekar’s Constant. Why is this number important? Perform the following process (called Kaprekar’s Routine):
- Take any two digit number whose digits are not all identical.
- Arrange the digits in descending and then ascending order to get two four digit numbers.
- Subtract the smaller number from the bigger number.
- 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.
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.
To read more, click here.
Click here for my previous post about Kaprekar. M x