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Trials and Tribulations

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The number 6174 is a really 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. Kaprekars operation In 1949 the mathematician D. R. Kaprekar from Devlali, India, devised a process now known as Kaprekars 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. It is a simple operation, but Kaprekar discovered it led to a surprising result. Lets try it out, starting with the number 2005, the digits of last year. The maximum number we can make with these digits is 5200, and the minimum is 0025 or 25 (if one or more of the digits is zero, embed these in the left hand side of the minimum number). The subtractions are: 5200 0025 = 5175 7551 1557 = 5994 9954 4599 = 5355 5553 3555 = 1998 9981 1899 = 8082 8820 0288 = 8532 8532 2358 = 6174 7641 1467 = 6174 When we reach 6174 the operation repeats itself, returning 6174 every time. We call the number 6174 a kernel of this operation. So 6174 is a kernel for Kaprekars operation, but is this as special as 6174 gets? Well not only is 6174 the only kernel for the operation, it also has one more surprise up its sleeve. Lets try again starting with a different number, say 1789. 9871 1789 = 8082 8820 0288 = 8532 8532 2358 = 6174 We reached 6174 again!
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