105 lines
7.1 KiB
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105 lines
7.1 KiB
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<h2>Kaprekar constant</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 414</h3><span style="width:300px;color:#666;">Published on Sunday, 10th February 2013, 07:00 am; Solved by 140</span></div>
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<div class="problem_content" role="problem">
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<P>
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6174 is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get when you sort the digits in decreasing order, we get 7641-1467=6174.<BR />
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Even more remarkable is that if we start from any 4 digit number and repeat this process of sorting and subtracting, we'll eventually end up with 6174 or immediately with 0 if all digits are equal.<BR />
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This also works with numbers that have less than 4 digits if we pad the number with leading zeroes until we have 4 digits.<BR />
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E.g. let's start with the number 0837:<BR />
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8730-0378=8352<BR />
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8532-2358=6174
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</P>
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<P>
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6174 is called the <b>Kaprekar constant</b>. The process of sorting and subtracting and repeating this until either 0 or the Kaprekar constant is reached is called the <b>Kaprekar routine</b>.
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</P>
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<P>
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We can consider the Kaprekar routine for other bases and number of digits.<BR />
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Unfortunately, it is not guaranteed a Kaprekar constant exists in all cases; either the routine can end up in a cycle for some input numbers or the constant the routine arrives at can be different for different input numbers.<BR >
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However, it can be shown that for 5 digits and a base b = 6t+3<img src='images/symbol_ne.gif' width='11' height='10' alt='≠' border='0' style='vertical-align:middle;' />9, a Kaprekar constant exists.<BR />
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E.g. base 15: (10,4,14,9,5)<sub>15</sub><BR />
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base 21: (14,6,20,13,7)<sub>21</sub>
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</P>
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<P>
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Define <var>C<sub>b</sub></var> to be the Kaprekar constant in base <var>b</var> for 5 digits.
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Define the function <var>sb(i)</var> to be
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<ul>
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<li> 0 if i = <var>C<sub>b</sub></var> or if <var>i</var> written in base <var>b</var> consists of 5 identical digits
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<li> the number of iterations it takes the Kaprekar routine in base <var>b</var> to arrive at <var>C<sub>b</sub></var>, otherwise
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</ul>
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Note that we can define <var>sb(i)</var> for all integers <var>i</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>b</var><sup>5</sup>. If <var>i</var> written in base <var>b</var> takes less than 5 digits, the number is padded with leading zero digits until we have 5 digits before applying the Kaprekar routine.
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</P>
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<P>
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Define <var>S(b)</var> as the sum of <var>sb(i)</var> for 0 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>i</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>b</var><sup>5</sup>.<BR >
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E.g. S(15) = 5274369<BR />
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S(111) = 400668930299
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</P>
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<P>
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Find the sum of S(6k+3) for 2 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> k <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 300.<BR />
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Give the last 18 digits as your answer.
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