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Number 6174 is really a mysterious number. At first glance, it might not seem so obvious. But as we are about to see, anyone who can subtract can uncover the mystery that makes 6174 so special. 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 the smallest number from the largest to get a new number;
• and carry on repeating the operation for each new number.

Let’s try it out, starting with the number 2021, the digits of this year. The maximum number we can make with these digits is 2210, and the minimum is 0122. The subtractions are :

\begin{aligned} 2210 - 0122 & = 2088 \\ 8820 - 0288 & = 8532 \\ 8532 - 2358 & = 6174 \\ 7641 - 1467 & = 6174 \end{aligned}

Extract from the article Mysterious 6174 from the website plus.maths.org

Questions

1. Try the Kaprekar’s operation on number 1234 and explain your calculations. What’s your conclusion?
2. Does the same phenomenon occur for three digit numbers? for five digit numbers?