6174 Number

[Floriana Grundy]

Therecan be analogous fixed points for digit lengths other than four; for instance, if we use 3-digit numbers, then most sequences (i.e., other than repdigits such as 111) will terminate in the value 495 in at most 6 iterations. Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Peyush constants" named after Peyush Dixit who solved this routine as a part of his IMO 2000 (International Mathematical Olympiad, Year 2000) thesis. [6]

In 1949 the mathematician D. R. Kaprekar from Devlali, India, devised a process now known as Kaprekar's operation. First choose a four digit number where the digits are not all the same (that is not 1111, 2222,...). Then rearrange the digits to get the largest and smallest numbers these digits can make. Finally, subtract thesmallest number from the largest to get a new number, and carry on repeating the operation for each new number.

It is a simple operation, but Kaprekar discovered it led to a surprising result. Let's try it out, starting with the number 2005, the digits of last year. The maximum number we can make with these digits is 5200, and the minimum is 0025 or 25 (if one or more of the digits is zero, embed these in the left hand side of the minimum number). The subtractions are:

When we reach 6174 the operation repeats itself, returning 6174 every time. We call the number 6174 a kernel of this operation. So 6174 is a kernel for Kaprekar's operation, but is this as special as 6174 gets? Well not only is 6174 the only kernel for the operation, it also has one more surprise up its sleeve. Let's try again starting with a different number, say 1789.

When we started with 2005 the process reached 6174 in seven steps, and for 1789 in three steps. In fact, you reach 6174 for all four digit numbers that don't have all the digits the same. It's marvellous, isn't it? Kaprekar's operation is so simple but uncovers such an interesting result. And this will become even more intriguing when we think about the reason why all four digit numbers reachthis mysterious number 6174.

The digits of any four digit number can be arranged into a maximum number by putting the digits in descending order, and a minimum number by putting them in ascending order. So for four digits a,b,c,d where

It was about 1975 when I first heard about the number 6174 from a friend, and I was very impressed at the time. I thought that it would be easy to prove why this phenomenon occurred but I could not actually find the reason why. I used a computer to check whether all four digit numbers reached the kernel 6174 in a limited number of steps. The program, which was about 50 statements in VisualBasic, checked all of 8991 four digit numbers from 1000 to 9999 where the digits were not all the same.

The table below shows the results: every four digit number where the digits aren't all equal reaches 6174 under Kaprekar's process, and in at most seven steps. If you do not reach 6174 after using Kaprekar's operation seven times, then you have made a mistake in your calculations and should try it again!

My computer program checked all 8991 numbers, but in his article Malcolm Lines explains that it is enough to check only 30 of all the possible four digit numbers when investigating Kaprekar's operation.

We can ignore the duplicates in Table 2 (the grey regions), and are left with just 30 numbers to follow through the rest of the process. The following figure shows the routes which these numbers take to reach 6174.

From this figure you can see how all the four digit numbers reach 6174 and reach it in at most seven steps. Even so I still think it is very mysterious. I guess Kaprekar, who discovered this number, was extremely clever or had a lot of time to think about it!

We have seen that four and three digit numbers reach a unique kernel, but how about other numbers? It turns out that the answers for those is not quite as impressive. Let try it out for a two digit number, say 28:

But what about five digits? Is there a kernel for five digit numbers like 6174 and 495? To answer this we would need to use a similar process as before: check the 120 combinations of a,b,c,d,e for ABCDE such that

Thankfully the calculations have already been done by a computer, and it is known that there is no kernel for Kaprekar's operation on five digit numbers. But all five digit numbers do reach one of the following three loops:

As Malcolm Lines points out in his article, it will take a lot of time to check what happens for six or more digits, and this work becomes extremely dull! To save you from this fate, the following table shows the kernels for two digit to ten digit numbers (for more see Mathews Archive ofRecreational Mathematics). It appears that Kaprekar's operation takes every number to a unique kernel only for three and four digit numbers.

We have seen that all three digit numbers reach 495, and all four digit numbers reach 6174 under Kaprekar's operation. But I have not explained why all such numbers reach a unique kernel. Is this phenomenon incidental, or is there some deeper mathematical reason why this happens? Beautiful and mysterious as the result is, it might just be incidental.

This is a very beautiful puzzle and you might think that a big mathematical theory should be hidden behind it. But in fact it's beauty is only incidental, there are other very similar, but not so beautiful, examples. Such as:

If I showed you Yamamoto's puzzle you would be inspired to solve it because it is so beautiful, but if I showed you the second puzzle you might not be interested at all. I think Kaprekar's problem is like Yamamoto's number guessing puzzle. We are drawn to both because they are so beautiful. And because they are so beautiful we feel there must be something more to them when in fact their beautymay just be incidental. Such misunderstandings have led to developments in mathematics and science in the past.

Is it enough to know all four digit numbers reach 6174 by Kaprekar's operation, but not know the reason why? So far, nobody has been able to say that all numbers reaching a unique kernel for three and four digit numbers is an incidental phenomenon. This property seems so surprising it leads us to expect that a big theorem in number theory hides behind it. If we can answer this question wecould find this is just a beautiful misunderstanding, but we hope not.

Yutaka Nishiyama is a professor at Osaka University of Economics, Japan. After studying mathematics at the University of Kyoto he went on to work for IBM Japan for 14 years. He is interested in the mathematics that occurs in daily life, and has written seven books about the subject. The most recent one, called "The mystery of five in nature", investigates, amongst other things, why manyflowers have five petals. Professor Nishiyama is currently visiting the University of Cambridge.

Yes I also learned basic in fact I learned MS-DOS and basic about the same time I later went on to try and understand C++. Let you in college but had difficulty with the arrays I probably should have studied Fortran first

I care. And people like me who were into programming in the 70s care. But the author explained his slip, above.

I taught math and programming through the 70s, 80s, and 90s. This is the first time I ran into this number pattern. Thanks for a fine presentation.

Very nice article.

For a non mathematician, I struggled for quite a while through the 4 lines of relations (e.g. trying to figure out how b had to be greater than c for the third one), it was only later that I read 'for those numbers where a>b>c>d'. Maybe that should go first?

Also am curious about how the iterations play out in non decimal bases?

Thanks!

3 is the only odd prime that is related to 9 by virtue of it being its divisor. All subsequent primes don't have any correlation with 9. Just the test on 11 will tell us by means of falsification if the hypothesis holds good.

Though mathematics is not my forte I found your artical interesting. As a Forester I was drawn to your the title of your book "The mystry of five in nature". My mind went to the biology term- 'Contact Parastikeits' (spelling may be off) which follows Fibonaccis sequence 1,1,2,3,5,8... Have you found the "Mystery" to follow this sequence?

Hello, when i was playing with numbers. i played with 2 digit numbers and i saw that always, or at least the many times i have done it. i came to a number that is a multypile of 9. is it true or i just got lucky?

The way I see it: imagine all the possible procedures you could do on all the possible intervals of numbers. You're looking for one reasonably simple procedure on an easily described interval which has the property that it has a unique fixed point. Loads of procedures have fixed points (e.g. Brouwer's theorem; $1-1/e \approx 63\%$ of all permutations are not derangements and therefore have at least one fixed point; etc.). Some of them will have unique fixed points. Finding a particularly nice procedure with a unique fixed point is probably going to be challenging, but when crowd-sourced it's not too surprising that somebody found one.

From that perspective, it's likely to be a fluke without any interesting proof. It'd be like looking through millions of rocks on a beach and trying to explain why the roundest one you found was so round. (That said, I'm completely ignorant of the specifics here, and for all I know there is a "nice" proof. That would be surprising, and I'd like to see it.)

Here's some hand-waving. You have a function on a finite set. Most functions are not one-to-one, so the image of the function is probably smaller than the original set. Now apply the function to the image. The image of the image is probably smaller than the image. If you keep doing this, you shouldn't be surprised that you arrive as some smaller fixed set that maps onto (and one-to-one) itself. For some functions, that final fixed set has only one element.

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