Converging to 495 – The Three Digit Kaprekar’s Constant

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.

Richard Courant

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

Photo Credit – Wikipedia

The number 495 is truly a strange number. At first go, it might not seem so obvious, but anyone who can subtract numbers can able to uncover the mystery that makes 495 so special.

To start with, choose a three-digit number. While choosing, please avoid choosing numbers with three identical digits like 111 or 999. For example, consider the number 201. Take the three 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 three digits, put some zeros to the left to maintain 3 digits. In the case of 201, the results are 210 and 012. Now take the difference between the largest (say L) and the smallest (say S). Thus we have 210 − 012 = 198. This type of operation is known as a Kaprekar operation.

D. R. Kaprekar discovered a special property of the number 495. The fact is, if the above operation will continue (means, if we will go on taking the difference between the largest and the smallest), ultimately the result will converge to the number 495, after a number of iterations. Considering the initial example of 201 (with S = 210 and L = 012), we have

210 − 012 = 198         [now S = 981 and L = 189]

981 − 189 = 792          [now S = 972 and L = 279]

972 − 279 = 693          [now S = 963 and L = 369]

963 − 369 = 594          [now S = 954 and L = 459]

954 − 459 = 495          [now S = 954 and L = 459]

954 − 459 = 495

Here, 495 is obtained after applying five Kaprekar operations.

Next consider the number 753. Taking the difference between the largest and the smallest successively as above, yields

753 − 357 = 396         [now S = 963 and L = 369]

963 − 369 = 594         [now S = 954 and L = 459]

954 − 459 = 495         [now S = 954 and L = 459]

954 − 459 = 495

Here, 495 is obtained after three Kaprekar operations.  Once the number becomes 495, the operation repeats, and 495 ultimately appears every time. No matter what the initial three-digit number may be, the sequence will eventually arrive at 495 and this works for all three digit numbers.

Can you find a number(s) that will take

(a) 1 step to reach 495?

(b) 2 steps to reach 495?

(c) 4 steps to reach 495?

(d) 6 steps to reach 495?

(e) more than 6 steps to reach 495?

There are 1000 three digit numbers, namely 100, 101, . . . , 999. Among them, nine are with all equal digits. Hence, there are 991 numbers with at least one different digit. In order to achieve 495 from any three 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 iteration required, 150 numbers with 1 iteration, 144 numbers with 2 iterations, 270 numbers with 3 iterations, 222 numbers with 4 iterations, 150 numbers with 5 iterations and 54 numbers with 6 iterations are there.

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call Math1089 – Mathematics for All!“.

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