## What is so Special about 6174?

What could possibly be special about a number like 6174; or perhaps we should say about 495 and 6174, because these two numbers are, in their own particular way, probably unique among all the decimal integers. Here we use the term ‘decimal integers’ because we are now going to look at digit patterns in numbers and these, of course, are dependent on the base to which we count. It is important to distinguish between the properties of numbers and of representations. The former (e.g. whether they are prime, perfect, amicable etc.) are absolute, while the latter depend on the counting base. Nevertheless, as we have seen with Benford in the previous chapter, a lot of fun can be had with representations as well as absolutes, and in this (and several of the following) chapters we shall look at the decimal representation in some detail. Whether as decimal representations, or as numbers in general, 495 and 6174 certainly do not exhibit any outward appearance of undue mystique; one is odd and one is even; they are not prime or perfect. In fact they give every indication of being rather ordinary, and in order to probe their particular brand of uniqueness it is necessary to go back to India and to the 1940’s when the Indian mathematician D. R. Kaprekar started to play with integers in a new and rather fascinating way. It was really a type of solitaire game in the beginning, and the rules are simple enough for anyone to play. In the era of the pocket calculator the procedure is not even particularly time consuming, and the moves of the game are as follows:

Think of an integer which contains more than one digit.

Arrange the digits in decreasing numerical order so as to make the largest number possible out of them (not all the digits must necessarily be different but they should not all be the same).

Arrange the digits a second time; this time in increasing numerical order so as to make the smallest possible number out of them (putting zeroes in the front if necessary).

Now subtract the smaller number (c) from the larger one (b) to give a new number with the same number of digits; that is including initial zeroes if necessary.

Finally, repeat the operations (b), (c), and (d) on the new number and continue.