90 lines
6.3 KiB
HTML
90 lines
6.3 KiB
HTML
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<title>Problem 186 - Project Euler</title>
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<h2>Connectedness of a network.</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 186</h3><span style="width:300px;color:#666;">Published on Saturday, 15th March 2008, 05:00 am; Solved by 1471</span></div>
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<p>Here are the records from a busy telephone system with one million users:</p>
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<div style='text-align:center;'>
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<table cellspacing='0' cellpadding='2' border='1' align='center'>
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<tr style='background-color:#c1daf9;'>
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<td>RecNr</td><td width='60' align='center'>Caller</td><td width='60' align='center'>Called</td></tr>
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<tr><td align='center'>1</td><td align='center'>200007</td><td align='center'>100053</td></tr>
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<tr><td align='center'>2</td><td align='center'>600183</td><td align='center'>500439</td></tr>
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<tr><td align='center'>3</td><td align='center'>600863</td><td align='center'>701497</td></tr>
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<tr><td align='center'>...</td><td align='center'>...</td><td align='center'>...</td></tr>
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</table>
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</div>
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<p>The telephone number of the caller and the called number in record n are Caller(n) = S<sub>2n-1</sub> and Called(n) = S<sub>2n</sub> where S<sub>1,2,3,...</sub> come from the "Lagged Fibonacci Generator":</p>
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<p>For 1 <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;' /> 55, S<sub>k</sub> = [100003 - 200003k + 300007k<sup>3</sup>] (modulo 1000000)<br />
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For 56 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> k, S<sub>k</sub> = [S<sub>k-24</sub> + S<sub>k-55</sub>] (modulo 1000000)</p>
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<p>If Caller(n) = Called(n) then the user is assumed to have misdialled and the call fails; otherwise the call is successful.</p>
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<p>From the start of the records, we say that any pair of users X and Y are friends if X calls Y or vice-versa. Similarly, X is a friend of a friend of Z if X is a friend of Y and Y is a friend of Z; and so on for longer chains.</p>
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<p>The Prime Minister's phone number is 524287. After how many successful calls, not counting misdials, will 99% of the users (including the PM) be a friend, or a friend of a friend etc., of the Prime Minister?</p>
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