WON plate135 | World!OfNumbers [ July 14, 2002 ] A very unique palindrome 549945 549945 is the only fixed-point palindrome for the Kaprekar process for base-10 numbers containing fewer than 16 digits. A neat palindromic fact spotted by Terry Trotter when he visited Walter Schneider's webpage at MATHEWS : Kaprekar Process ( now alas a broken link! ) This Kaprekar process is illustrated as follows : 549945 995544 – 445599 549945 [ July 19, 2002 ] No person is better placed to explain the topic than Terry Trotter (email) himself.   A Palindrome à la Kaprekar   Here is an interesting palindrome: 995544445599. To see why, we must first introduce a popular idea that has fascinated mathematicians and school students alike for sometime. It's called Kaprekar's Ordered Subtraction Operation (OSO). The OSO may be described thus: Start with a four-digit number whose digits are not all equal, arrange the digits in descending and ascending order, subtract and repeat the process. Then the process terminates on the number 6174 after seven or fewer steps. The amazing thing here is that 6174 always results, no matter what the initial 4-digit number might be (excepting those consisting of only one digit, like 3333, of course). As one does the OSO process with numbers of more or fewer digits, different outcomes can result. With 3-digit numbers, the constant final outcome is always 495. I call that a terminator. With 2-digit numbers, the following five-term cycle is the only outcome 09 81 63 27 45 09 But things really become interesting when using numbers of 5 or more digits. Here cycles and terminators can occur for a given case. (See Walter Schneider's website for all the outcomes up to 15-digit numbers in base 10.) A quick examination of that table reveals the fact that there is only one palindrome. It occurs for 6-digit numbers. 549945 is one of two terminators possible, accompanied by one cycle of seven terms. By now it should be obvious how the 12-digit palindrome above was created. It is a number with the interesting property that can be stated as: A palindrome with an even number of digits such that if it is separated into 'halves', the positive difference of the two parts is itself a palindrome. 995544445599 995544 – 445599 549945. Now, I can only but wonder if there are other palindromes that share this special property. ps1. regarding your doubt about fixed points... I just call such things terminators or self-producers. 6174 is often called Kaprekar's Constant. ps2. For every Kaprekar Palindrome, as we might call those numbers with the split-half property, there is always a companion. Just reverse the 'halves', like so: 445599995544 trivial, perhaps, but necessary for completeness. A000135 Prime Curios! Prime Puzzle Wikipedia 135 Le nombre 135
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