“A drunk wants to continue drinking wine to maintain that pleasant state. The same applies to me when it comes to numbers”, said the discoverer of the phenomenon that has intrigued mathematicians since 1949.
The number 6174 appears at first glance to be nothing special, but it has intrigued mathematicians and number theory enthusiasts since 1949. Why?
Let’s get straight to the point. To understand, follow the following steps:
1. Choose any four-digit number that contains at least two different digits, including zero (for example, 1234).
2. Arrange the digits in descending order (which following this example would be 4321).
3. Now arrange the digits in ascending order: 1234
4. Subtract the smallest number from the largest: 4321 – 1234 = 3087
5. Now repeat the last three steps.
First, we organize the digits in descending order: 8730. Then, in ascending order: 0378. We subtract the smallest from the largest: 8730 – 0378 = 8352.
Again, we rearrange the digits and subtract: 8532 – 2358 = 6174. And once again, we rearrange the digits and subtract: 7641 – 1467 = 6174.
From here, there is no point continuing, as we would repeat the same operation.
Let’s test 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 doesn’t matter what number you start with: you always arrive at 6174.
A lover of numbers
This curious phenomenon is known as the Kaprekar constantnamed after the person who discovered the mysterious beauty of the number 6174 and presented it at the Madras Mathematical Conference in 1949: Dattatreya Ramchandra Kaprekar (1905-1986), a self-confessed lover of number theory.
“A drunk wants to keep drinking wine to stay in that pleasant state. The same applies to me when it comes to numbers,” he used to say.
Kaprekar was a teacher in a small Indian town called Devlali or Deolali and was often invited to speak at other schools about his unique methods and fascinating numerical observations. However, several Indian mathematicians laughed at his ideas.
Perhaps they were in fact trivial: it is certain that, although Kaprekar’s Constant is surprising and leads us to suspect that behind it there is a great theorem, So far, nothing has been revealed.
Who laughs last…
But not everything needs to be useful to be fun and interesting. Kaprekar became well-known within and outside India because many other mathematicians found his ideas intriguing and, like him, continued to play with numbers.
Yutaka Nishiyama, from the Osaka University of Economics, in Japan, states in +plus magazine that he used a computer to check whether there was a limited number of steps to reach 6174. He concluded that the maximum number of steps is 7, that is, if you do not reach 6174 after seven operations, there was an error in the calculations and you should try again.
In other investigations, it was discovered that the same phenomenon occurs when, instead of starting with four digits, you start with three.
Let’s try with 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 doesn’t happen in all cases: only with three or four digit numbers (at least 2 to 10 digits, which is what was tested).
To encourage students
Currently, the non-profit organization Scigram Technologies Foundation is developing a computer teaching platform in India especially aimed at rural and tribal schools. The organization transformed the number 6174 into a colorful table that illustrates this report.
Co-founder Girish Arabale explains that they always seek to inspire and motivate school-age children who tend to hate math.
“The Kaprekar Constant 6174 is one of those beautiful numbers, and the steps leading to its discovery create an ‘aha!’ moment, the kind that is sorely missed in traditional mathematics curricula.”
As seen below, they assigned a color to each number of steps needed to reach 6174 (remember that there is a maximum of 7 steps).
A code was then written that can be easily recreated on a Raspberry Pi, a cheap computer often used to teach the Wolfram language, available for free on the Raspberry Pi. A program thus generated patterns with the steps that lead to the number 6174 for each of the 10 thousand existing 4-digit numbers, creating the table below with different colors.
Recreational mathematics
Kaprekar’s Constant was not the only fruit of the Indian’s passion for numbers. Among his collection of ideas is also the Kaprekar number.
It is a number with the interesting property that, if it is squared and the two equal parts of the result are added, the original number is obtained. For example:
297² = 88.209
88 + 209 = 297
Other examples of Kaprekar Numbers are: 9, 45, 55, 703, 17,344, 538,461… test and check! But be careful: when dividing the number into parts that are added, leave the longest part on the right (in the example, when dividing 88,209 into two groups, 88 and 209 are obtained and not 882 and 09).
Good calculations!
