83 lines
6.1 KiB
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83 lines
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<h2>Special subset sums: optimum</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 103</h3><span style="width:300px;color:#666;">Published on Friday, 26th August 2005, 06:00 pm; Solved by 3613</span></div>
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<p>Let S(A) represent the sum of elements in set A of size <i>n</i>. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are true:</p>
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<ol style="list-style-type:lower-roman;">
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<li>S(B) <img src='images/symbol_ne.gif' width='11' height='10' alt='≠' border='0' style='vertical-align:middle;' /> S(C); that is, sums of subsets cannot be equal.</li>
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<li>If B contains more elements than C then S(B) <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> S(C).</li>
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</ol>
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<p>If S(A) is minimised for a given <i>n</i>, we shall call it an optimum special sum set. The first five optimum special sum sets are given below.</p>
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<p style="margin-left:50px;"><i>n</i> = 1: {1}<br />
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<i>n</i> = 2: {1, 2}<br />
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<i>n</i> = 3: {2, 3, 4}<br />
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<i>n</i> = 4: {3, 5, 6, 7}<br />
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<i>n</i> = 5: {6, 9, 11, 12, 13}</p>
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<p>It <i>seems</i> that for a given optimum set, A = {<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ... , <i>a</i><sub>n</sub>}, the next optimum set is of the form B = {<i>b</i>, <i>a</i><sub>1</sub>+<i>b</i>, <i>a</i><sub>2</sub>+<i>b</i>, ... ,<i>a</i><sub>n</sub>+<i>b</i>}, where <i>b</i> is the "middle" element on the previous row.</p>
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<p>By applying this "rule" we would expect the optimum set for <i>n</i> = 6 to be A = {11, 17, 20, 22, 23, 24}, with S(A) = 117. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for <i>n</i> = 6 is A = {11, 18, 19, 20, 22, 25}, with S(A) = 115 and corresponding set string: 111819202225.</p>
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<p>Given that A is an optimum special sum set for <i>n</i> = 7, find its set string.</p>
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<p class="info">NOTE: This problem is related to problems <a href="index.php?section=problems&id=105">105</a> and <a href="index.php?section=problems&id=106">106</a>.</p>
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