186 lines
7.5 KiB
HTML
186 lines
7.5 KiB
HTML
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<title>Problem 298 - Project Euler</title>
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<h2>Selective Amnesia</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 298</h3><span style="width:300px;color:#666;">Published on Friday, 25th June 2010, 09:00 pm; Solved by 426</span></div>
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<div class="problem_content" role="problem">
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table.p298, table.p298 th, table.p298 td {
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border-color: black black black black;
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table.p298 {
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background-color: rgb(255,255,255);
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}
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table.p298 th, table.p298 td {
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padding: 1px 6px 1px 6px;
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}
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table.p298 th { background-color: rgb(200,220,250); }
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table.p298 td { background-color: rgb(255,255,255); }
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</style>
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<p>Larry and Robin play a memory game involving of a sequence of random numbers between 1 and 10, inclusive, that are called out one at a time. Each player can remember up to 5 previous numbers. When the called number is in a player's memory, that player is awarded a point. If it's not, the player adds the called number to his memory, removing another number if his memory is full.</p>
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<p>Both players start with empty memories. Both players always add new missed numbers to their memory but use a different strategy in deciding which number to remove:<br />
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Larry's strategy is to remove the number that hasn't been called in the longest time.<br />
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Robin's strategy is to remove the number that's been in the memory the longest time.</p>
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<p>Example game:<br />
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<center><table class="p298">
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<tr>
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<th>Turn</th>
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<th>Called<br />number</th>
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<th style="text-align:right">Larry's<br />memory</th>
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<th>Larry's<br />score</th>
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<th style="text-align:right">Robin's<br />memory</th>
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<th>Robin's<br />score</th>
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</tr>
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<tr>
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<td>1</td>
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<td>1</td>
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<td style="text-align:right">1</td>
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<td>0</td>
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<td style="text-align:right">1</td>
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<td>0</td>
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</tr>
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<tr>
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<td>2</td>
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<td>2</td>
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<td style="text-align:right">1,2</td>
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<td>0</td>
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<td style="text-align:right">1,2</td>
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<td>0</td>
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</tr>
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<tr>
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<td>3</td>
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<td>4</td>
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<td style="text-align:right">1,2,4</td>
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<td>0</td>
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<td style="text-align:right">1,2,4</td>
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<td>0</td>
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</tr>
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<tr>
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<td>4</td>
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<td>6</td>
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<td style="text-align:right">1,2,4,6</td>
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<td>0</td>
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<td style="text-align:right">1,2,4,6</td>
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<td>0</td>
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</tr>
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<tr>
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<td>5</td>
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<td>1</td>
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<td style="text-align:right">1,2,4,6</td>
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<td>1</td>
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<td style="text-align:right">1,2,4,6</td>
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<td>1</td>
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</tr>
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<tr>
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<td>6</td>
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<td>8</td>
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<td style="text-align:right">1,2,4,6,8</td>
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<td>1</td>
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<td style="text-align:right">1,2,4,6,8</td>
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<td>1</td>
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</tr>
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<tr>
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<td>7</td>
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<td>10</td>
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<td style="text-align:right">1,4,6,8,10</td>
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<td>1</td>
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<td style="text-align:right">2,4,6,8,10</td>
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<td>1</td>
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</tr>
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<tr>
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<td>8</td>
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<td>2</td>
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<td style="text-align:right">1,2,6,8,10</td>
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<td>1</td>
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<td style="text-align:right">2,4,6,8,10</td>
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<td>2</td>
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</tr>
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<tr>
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<td>9</td>
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<td>4</td>
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<td style="text-align:right">1,2,4,8,10</td>
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<td>1</td>
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<td style="text-align:right">2,4,6,8,10</td>
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<td>3</td>
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</tr>
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<tr>
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<td>10</td>
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<td>1</td>
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<td style="text-align:right">1,2,4,8,10</td>
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<td>2</td>
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<td style="text-align:right">1,4,6,8,10</td>
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<td>3</td>
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</tr></table></center></p>
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<p>Denoting Larry's score by <var>L</var> and Robin's score by <var>R</var>, what is the expected value of |<var>L</var>-<var>R</var>| after 50 turns? Give your answer rounded to eight decimal places using the format x.xxxxxxxx .</p>
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