77 lines
5.5 KiB
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77 lines
5.5 KiB
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<h2>Special subset sums: meta-testing</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 106</h3><span style="width:300px;color:#666;">Published on Friday, 7th October 2005, 06:00 pm; Solved by 2919</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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<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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<p>For this problem we shall assume that a given set contains <i>n</i> strictly increasing elements and it already satisfies the second rule.</p>
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<p>Surprisingly, out of the 25 possible subset pairs that can be obtained from a set for which <i>n</i> = 4, only 1 of these pairs need to be tested for equality (first rule). Similarly, when <i>n</i> = 7, only 70 out of the 966 subset pairs need to be tested.</p>
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<p>For <i>n</i> = 12, how many of the 261625 subset pairs that can be obtained need to be tested for equality?</p>
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<p class="info">NOTE: This problem is related to problems <a href="index.php?section=problems&id=103">103</a> and <a href="index.php?section=problems&id=105">105</a>.</p>
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