157 lines
7.3 KiB
HTML
157 lines
7.3 KiB
HTML
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<title>Problem 334 - Project Euler</title>
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<h2>Spilling the beans</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 334</h3><span style="width:300px;color:#666;">Published on Saturday, 23rd April 2011, 04:00 pm; Solved by 301</span></div>
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<div class="problem_content" role="problem">
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<p>In Plato's heaven, there exist an infinite number of bowls in a straight line.<br />
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Each bowl either contains some or none of a finite number of beans.<br />
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A child plays a game, which allows only one kind of move: removing two beans from any bowl, and putting one in each of the two adjacent bowls.<br /> The game ends when each bowl contains either one or no beans.</p>
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<p>For example, consider two adjacent bowls containing 2 and 3 beans respectively, all other bowls being empty. The following eight moves will finish the game:</p>
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<div align='center'><img src="project/images/p_334_beans.gif" /></div>
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<p>You are given the following sequences:<br />
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<table class="formula" style="margin-left:50px;">
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<tr>
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<td>
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<var>t</var><sub><i>0</i></sub> = 123456.
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</td>
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</tr></table>
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<table class="formula" style="margin-left:50px;">
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<tr>
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<td>
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<var>t</var><sub><i>i</i></sub> =
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</td>
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<td><img src="project/images/p_334_cases.gif" /></td>
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<td>
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<table class="formula">
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<tr>
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<td></td>
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<td>
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<table class="frac">
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<tr><td><var>t</var><sub><i>i-1</i></sub></td></tr>
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<tr><td class="overline">2</td></tr>
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</table>
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</td>
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<td>
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,
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</td>
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<td></td>
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<td>
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if <var>t</var><sub><i>i-1</i></sub> is even
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</td>
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</tr>
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<tr>
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<td><img src="project/images/p_334_lfloor.gif" /></td>
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<td>
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<table class="frac">
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<tr><td><var>t</var><sub><i>i-1</i></sub></td></tr>
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<tr><td class="overline">2</td></tr>
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</table>
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</td>
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<td>
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<img src="project/images/p_334_rfloor.gif" />
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</td>
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<td>
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926252,
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</td>
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<td>
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if <var>t</var><sub><i>i-1</i></sub> is odd
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</td>
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</tr>
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</table>
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<td>
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</tr>
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<tr>
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<td></td>
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<td></td>
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<td>
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where <img src='images/symbol_lfloor.gif' width='6' height='16' alt='⌊' border='0' style='vertical-align:middle;' /><var>x</var><img src='images/symbol_rfloor.gif' width='6' height='16' alt='⌋' border='0' style='vertical-align:middle;' /> is the floor function
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</td>
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</tr>
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<tr>
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<td></td>
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<td></td>
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<td>
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and <img src="project/images/p_334_oplus.gif" /> is the bitwise XOR operator.
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</td>
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</tr></table>
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<table class="formula" style="margin-left:50px;">
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<tr>
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<td>
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<var>b</var><sub><i>i</i></sub> = ( <var>t</var><sub><i>i</i></sub> mod 2<sup>11</sup>) + 1.
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</td>
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</tr></table></p>
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<p>The first two terms of the last sequence are <var>b</var><sub><i>1</i></sub> = 289 and <var>b</var><sub><i>2</i></sub> = 145.<br />
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If we start with <var>b</var><sub><i>1</i></sub> and <var>b</var><sub><i>2</i></sub> beans in two adjacent bowls, 3419100 moves would be required to finish the game.</p>
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<p>Consider now 1500 adjacent bowls containing <var>b</var><sub><i>1</i></sub>, <var>b</var><sub><i>2</i></sub>,..., <var>b</var><sub><i>1500</i></sub> beans respectively, all other bowls being empty. Find how many moves it takes before the game ends.</p>
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