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puzzles/project_euler/pe/p315.html
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puzzles/project_euler/pe/p315.html
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<!DOCTYPE html>
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<html lang="en">
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<meta name="author" content="Colin Hughes" />
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<title>Problem 315 - Project Euler</title>
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<h2>Digital root clocks</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 315</h3><span style="width:300px;color:#666;">Published on Sunday, 19th December 2010, 10:00 am; Solved by 1251</span></div>
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<div class="problem_content" role="problem">
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<p><div align='center'><img src="project/images/p_315_clocks.gif" /></div></p>
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<p>Sam and Max are asked to transform two digital clocks into two "digital root" clocks.<br />
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A digital root clock is a digital clock that calculates digital roots step by step.</p>
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<p>When a clock is fed a number, it will show it and then it will start the calculation, showing all the intermediate values until it gets to the result.<br />
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For example, if the clock is fed the number 137, it will show: "<b>137</b>" <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> "<b>11</b>" <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> "<b>2</b>" and then it will go black, waiting for the next number.</p>
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<p>Every digital number consists of some light segments: three horizontal (top, middle, bottom) and four vertical (top-left, top-right, bottom-left, bottom-right).<br />
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Number "<b>1</b>" is made of vertical top-right and bottom-right, number "<b>4</b>" is made by middle horizontal and vertical top-left, top-right and bottom-right. Number "<b>8</b>" lights them all.</p>
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<p>The clocks consume energy only when segments are turned on/off.<br />
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To turn on a "<b>2</b>" will cost 5 transitions, while a "<b>7</b>" will cost only 4 transitions.</p>
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<p>Sam and Max built two different clocks.</p>
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<p>Sam's clock is fed e.g. number 137: the clock shows "<b>137</b>", then the panel is turned off, then the next number ("<b>11</b>") is turned on, then the panel is turned off again and finally the last number ("<b>2</b>") is turned on and, after some time, off.<br />
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For the example, with number 137, Sam's clock requires:<br />
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<table><tr>
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<td>"<b>137</b>"</td>
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<td>:</td>
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<td>(2 + 5 + 4) <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 2 = 22 transitions ("<b>137</b>" on/off).</td>
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</tr><tr>
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<td>"<b>11</b>"</td>
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<td>:</td>
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<td>(2 + 2) <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 2 = 8 transitions ("<b>11</b>" on/off).</td>
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</tr><tr>
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<td>"<b>2</b>"</td>
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<td>:</td>
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<td>(5) <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 2 = 10 transitions ("<b>2</b>" on/off).</td>
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</tr></table>
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For a grand total of 40 transitions.</p>
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<p>Max's clock works differently. Instead of turning off the whole panel, it is smart enough to turn off only those segments that won't be needed for the next number.<br />
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For number 137, Max's clock requires:<br />
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<table><tr>
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<td>"<b>137</b>"<br /><br /></td>
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<td>:<br /><br /></td>
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<td>2 + 5 + 4 = 11 transitions ("<b>137</b>" on)<br />
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7 transitions (to turn off the segments that are not needed for number "<b>11</b>").</td>
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</tr><tr>
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<td>"<b>11</b>"<br /><br /><br /></td>
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<td>:<br /><br /><br /></td>
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<td>0 transitions (number "<b>11</b>" is already turned on correctly)<br />
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3 transitions (to turn off the first "<b>1</b>" and the bottom part of the second "<b>1</b>"; <br />
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the top part is common with number "<b>2</b>").</td>
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</tr><tr>
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<td>"<b>2</b>"<br /><br /></td>
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<td>:<br /><br /></td>
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<td>4 tansitions (to turn on the remaining segments in order to get a "<b>2</b>")<br />
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5 transitions (to turn off number "<b>2</b>").</td>
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</tr></table>
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For a grand total of 30 transitions.</p>
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<p>Of course, Max's clock consumes less power than Sam's one.<br />
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The two clocks are fed all the prime numbers between A = 10<sup>7</sup> and B = 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />10<sup>7</sup>. <br />
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Find the difference between the total number of transitions needed by Sam's clock and that needed by Max's one.</p>
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