The mysterious number 6174 that has intrigued mathematicians for 70 years

6174 looks like any number, out of the air, without mention of glory. However, he has intrigued mathematicians and lovers of number theory since 1949.

Because? Well, look at this so curious.

1. Choose a four-digit number consisting of at least two different digits, including zero, for example, 1234.

2. Arrange the numbers in descending order, which in our example would be 4321.

3. Now, arrange the number in ascending order: 1234

4. Subtract the smallest number from as many: 4321 – 1234

5. And now, repeat the last three steps

We are going to do it:

4321 – 1234 = 3087

Then we organize the digits of 3087 in descending order and the remains of 8730, and in ascending order, 0378, and subtract:

8730 – 0378 = 8352

again, we organize the numbers for result 8352 and subtract them:

8532 – 2358 = 6174

Again, in decreasing order -7641- and increasing -1467-, subtract:

7641 – 1467 = 6174

As you will see, it is now useless to follow, because we would only repeat the same operation.

Let's deal with another number. How about 2005?

5200 – 0025 = 5175

7551 – 1557 = 5994

9954 – 4599 = 5355

5553 – 3555 = 1998

9981 – 1899 = 8082

8820 – 0288 = 8532

8532 – 2358 = 6174

7641 – 1467 = 6174

It turns out that the number with which you start does not matter, you always reach 6174 and then, the operation is repeated, with the same result again and again: 6174.

This is what is called Kaprekar's constant, the one who discovered the mysterious beauty of 6174 and presented it at the mathematical conference of Madras in 1949: Dattatreya Ramchandra Kaprekar (1905-1986), a drug addict confessed to number theory.

"A drunkard wants to keep drinking wine to stay in this pleasant state, the same goes for me when it comes to numbers," he used to say.

Kaprekar was a teacher in a small Indian population called Devlali or Deolali and was often invited to speak in other schools about his unique methods and fascinating digital observations.

However, many Indian mathematicians laughed at his ideas, calling them trivial.

Perhaps they are: it should be noted that even though it is so surprising that it leads us to believe that it hides a great theorem of number theory, at least so far, the Kaprekar constant did not reveal anything like that.

Since not everything needs to be useful to be attractive, fun and interesting, Kaprekar has made itself known both inside and outside of India, many other mathematicians having found their intriguing ideas.

And like him, they continued to play with their numbers.

Yutaka Nishiyama, from the Osaka University of Economics in Japan, for example, says in the magazine + more than he used a computer to see if there were a limited number of steps to reach 6174.

After checking, he established that the maximum number of steps was 7, that is, if you do not reach 6174 after using the Kaprekar operation seven times, you made a mistake in your calculations and you have to try again.

In other explorations, it has been discovered that the same phenomenon occurs when, instead of starting with four-digit numbers, you start with those of three.

Let's try number 574:

754 – 457 = 297

972 – 279 = 693

963 – 369 = 594

954 – 459 = 495

954 – 459 = 495

As you can see, the magic number in this case is 495.

And no, this does not happen in other cases: only when you start with three or four digit numbers (at least 2 to 10 digits, which has been checked).

Meanwhile, in India, the non-profit Scigram Technologies Foundation, which is developing an "IT learning platform" specifically for rural and tribal schools, has produced 6174 and has transformed it into three colorful illustrations which adorn this article from here on.

Co-founder Girish Arabale explained to BBC Mundo that what they are always looking for, is to inspire especially schoolchildren who generally hate mathematics, showing them what they are calling " Aha moments! " to motivate them.

"The constant Kaprekar 6174 is one of those beautiful numbers and the steps leading to its discovery create a moment" Aha! "This is usually missing in traditional math programs."

Then, they badigned a color to each number of steps required to reach 6174 (remember there was a maximum of 7 steps):

… wrote a code that can be easily recreated on a Raspberry Pi (a popular tool in STEM teaching) in a Wolfram language, available for free on Raspberry Pi …

… and execute the program for each of the existing 10,000 4-digit numbers, creating patterns whose steps leading to the number 6174 are arranged in a grid with the different colors.

Kaprekar's constant was not the only contribution of this pbadion for numbers to recreational mathematics.

The number of Kaprekar is also part of his collection of ideas.

This is a number with the interesting property that it's square, adding two equal parts of the result gives you the original number. This operation is Kaprekar operation.

To clarify, an example.

297² = 88.209 and 88 + 209 = 297

Some additional examples of Kaprekar numbers are: 9, 45, 55, 703, 17, 344, 538, 461 … try them and see!

Remember: when you divide the number you want to add the parts, leave the longest part to the right (in the example, if you divide the 88209 in two, you have two groups: one with two digits and the other with three, so indications, when separated it stays 88 and 209 and not 882 and 09).