Let’s play with 6174 – The four digit Kaprekar’s Constant

First, it is necessary to study the facts, to multiply the number of observations, and then later to search for formulas that connect them so as thus to discern the particular laws governing a certain class of phenomena. In general, it is not until after these particular laws have been established that one can expect to discover and articulate the more general laws that complete theories by bringing a multitude of apparently very diverse phenomena together under a single governing principle.

Augustin Louis Cauchy

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Dattatreya Ramchandra Kaprekar was an well known Indian recreational mathematician. The famous Kaprekar’s constant named after him.

The four-digit number 6174 is truly a strange number, not so obvious at first go. But anyone capable of subtracting numbers can able to uncover the mystery that makes 6174 so special.

To start with, choose a four-digit number. While choosing, please avoid choosing numbers with four identical digits like 3333 or 4444. Take the four digits that made up the number and construct the largest and smallest possible numbers from them. By chance, if the number formed contains less than four digits, put some zeros to the left to maintain 4 digits.

For example, consider the current year 2021. In the case of 2021, the results are 2210 and 0122. Now take the difference between the largest (say L) and the smallest (say S). Thus we have 2210 − 0122 = 2088. This type of operation is known as a Kaprekar operation.

D. R. Kaprekar discovered a special property of the number 6174. The fact is, if the above operation will continue (that is, if we will go on taking the difference between the larger and smaller), ultimately the result will converge to the number 6174, after a number of iterations. Considering the initial example of 2021 (with S = 2210 and L = 0122), we have

2210 − 0122 = 2088   [now S = 8820 and L = 0288]

8820 − 0288 = 8532   [now S = 8532 and L = 2358]

8532 − 2358 = 6174    [now S = 7641 and L = 1467]

7641 − 1467 = 6174

Here, 6174 is obtained after applying three Kaprekar operations. Next consider the number 2005. Taking the difference between the largest and the smallest successively as above, yields

5200 − 0025 = 5175   [now S = 7551 and L = 1557]

7551 − 1557 = 5994   [now S = 9954 and L = 4599]

9954 − 4599 = 5355   [now S = 5553 and L = 3555]

5553 − 3555 = 1998   [now S = 9981 and L = 1899]

9981 − 1899 = 8082   [now S = 8820 and L = 0288]

8820 − 0288 = 8532   [now S = 8532 and L = 2358]

8532 − 2358 = 6174   [now S = 7641 and L = 1467]

7641 − 1467 = 6174

Here, 6174 is obtained after seven Kaprekar operations.  Once the number becomes 6174, the operation repeats and 6174 ultimately appears every time. No matter what the initial four-digit number may be, the sequence will eventually arrive at 6174 and this works for all four digit numbers.

Can you find a number(s) that will take

(a) 1 step to reach 6174?

(b) 2 steps to reach 6174?

(c) 4 steps to reach 6174?

(d) 5 steps to reach 6174?

(e) 6 steps to reach 6174?

(f) more than 7 steps to reach 6174?

There are 9000 four digit numbers, namely 1000, 1001, . . . , 9999. Among them, nine are with all equal digits. Hence, there are 8991 numbers with at least one different digits. In order to achieve 6174 from any four digit number, it’s important to know how many iterations required and if the number of iterations are given, then we should know how many such numbers are there?

There is only 1 number with 0 iterations required to obtain 6174, 356 numbers with 1 iteration, 519 numbers with 2 iterations, 2124 numbers with 3 iterations, 1124 numbers with 4 iterations, 1379 numbers with 5 iterations, 1508 numbers with 6 iterations and 1980 numbers are there with 7 iterations required to reach 6174.

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