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puzzles/project_euler/pe/p255.html
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<h2>Rounded Square Roots</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 255</h3><span style="width:300px;color:#666;">Published on Friday, 11th September 2009, 09:00 pm; Solved by 561</span></div>
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<div class="problem_content" role="problem">
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<p>We define the <i>rounded-square-root</i> of a positive integer <var>n</var> as the square root of <var>n</var> rounded to the nearest integer.</p>
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<p>The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of <var>n</var>:</p>
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<p>Let <var>d</var> be the number of digits of the number <var>n</var>.<br />
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If <var>d</var> is odd, set <var>x</var><sub>0</sub> = 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />10<sup>(<var>d</var>-1)⁄2</sup>.<br />
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If <var>d</var> is even, set <var>x</var><sub>0</sub> = 7<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />10<sup>(<var>d</var>-2)⁄2</sup>.<br />
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Repeat:</p>
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<img src="project/images/p_255_Heron.gif" /></p>
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<!--
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<table align='center'>
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<tr><td><var>x</var><sub><var>k</var>+1</sub> =</td>
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<td style='font-size:220%'>⌊</td>
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<td style='text-align:center;'><var>x</var><sub><var>k</var></sub> + <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' /><var>n</var>⁄<var>x</var><sub><var>k</var></sub><img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' /><br />
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<img src='images/blackdot.gif' width='75' height='1' alt='' /><br />
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2</td><td><td style='font-size:220%'>⌋</td></tr>
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</table> -->
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<p>until <var>x</var><sub><var>k</var>+1</sub> = <var>x</var><sub><var>k</var></sub>.
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</p>
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<p>As an example, let us find the rounded-square-root of <var>n</var> = 4321.<br />
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<var>n</var> has 4 digits, so <var>x</var><sub>0</sub> = 7<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />10<sup>(4-2)⁄2</sup> = 70.<br />
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<img src='project/images/p_255_Example.gif'>
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<!--<var>x</var><sub>1</sub> = <img src='images/symbol_lfloor.gif' width='6' height='16' alt='⌊' border='0' style='vertical-align:middle;' />(70 + <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' />4321⁄70<img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' />)⁄2<img src='images/symbol_rfloor.gif' width='6' height='16' alt='⌋' border='0' style='vertical-align:middle;' /> = 66.<br />
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<var>x</var><sub>2</sub> = <img src='images/symbol_lfloor.gif' width='6' height='16' alt='⌊' border='0' style='vertical-align:middle;' />(66 + <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' />4321⁄66<img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' />)⁄2<img src='images/symbol_rfloor.gif' width='6' height='16' alt='⌋' border='0' style='vertical-align:middle;' /> = 66.--><br />
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Since <var>x</var><sub>2</sub> = <var>x</var><sub>1</sub>, we stop here.<br />
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So, after just two iterations, we have found that the rounded-square-root of 4321 is 66 (the actual square root is 65.7343137…).
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</p>
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<p>The number of iterations required when using this method is surprisingly low.<br />
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For example, we can find the rounded-square-root of a 5-digit integer (10,000 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 99,999) with an average of 3.2102888889 iterations (the average value was rounded to 10 decimal places).
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</p>
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<p>Using the procedure described above, what is the average number of iterations required to find the rounded-square-root of a 14-digit number (10<sup>13</sup> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 10<sup>14</sup>)?<br />
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Give your answer rounded to 10 decimal places.
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</p>
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<p>Note: The symbols <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;' /> and <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' /><var>x</var><img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' /> represent the <dfn title='the largest integer not greater than x'>floor function</dfn> and <dfn title='the smallest integer not less than x'>ceiling function</dfn> respectively.
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