<!DOCTYPE html> <html lang="en"> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <meta charset="utf-8"> <title>Problem 353 - Project Euler</title> <link rel="shortcut icon" href="favicon.ico"> <link rel="stylesheet" type="text/css" href="pe_template_style.css"> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ jax: ["input/TeX", "output/HTML-CSS"], tex2jax: { inlineMath: [ ["$","$"], ["\\(","\\)"] ], displayMath: [ ["$$","$$"], ["\\[","\\]"] ], processEscapes: true }, "HTML-CSS": { availableFonts: ["TeX"] } }); </script> <script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML"> </script> <script type="text/javascript" src="md5.min.js"></script> <script type="text/javascript"> function check_ans(ans) { if (md5(ans) === "211b5626459b71bafc78478d18bdc3") { alert("Correct!"); } else { alert("Sorry, that is incorrect!"); }; }; </script> </head> <body> <div id="container"> <div id="nav" class="noprint"> <ul> <li><a href="about.html" title="About">About</a></li> <li id="current"><a href="problems.html" title="Problems">Problems</a></li> <li><a href="news.html" title="News">News</a></li> <li><a href="http://forum.projecteuler.net" title="Project Euler Forum">Forum</a></li> </ul> <div style="text-align:right;padding:30px 10px 0 0;"> <a href="rss2_euler.xml"><img src="images/icon_rss.png" alt="RSS Feed" title="RSS Feed"></a> </div> </div> <div id="logo" class="noprint"> <img src="images/pe_banner.png" alt="Project Euler .net"> </div> <div id="content"> <div id="notification">This website is a semi-functional mirror of the original <a href="https://projecteuler.net/">Project Euler</a>. More information is available on <a href="https://github.com/wackyapples/ProjectEulerMirror/blob/gh-pages/README.md">GitHub</a>.</div> <div style="text-align:center;" class="print"><img src="images/pe_banner.png" alt="projecteuler.net" style="border:none;"></div> <h2>Risky moon</h2> <h3>Problem 353</h3> <div style="font-size:80%;color:#999;">Published on 09 October 2011 at 04:00 am [Server Time]</div> <div class="problem_content" role="problem"> <p> A moon could be described by the sphere C(<var>r</var>) with centre (0,0,0) and radius <var>r</var>. </p> <p> There are stations on the moon at the points on the surface of C(<var>r</var>) with integer coordinates. The station at (0,0,<var>r</var>) is called North Pole station, the station at (0,0,-<var>r</var>) is called South Pole station. </p> <p> All stations are connected with each other via the shortest road on the great arc through the stations. A journey between two stations is risky. If <var>d</var> is the length of the road between two stations, (<var>d</var>/(π <var>r</var>))<sup>2</sup> is a measure for the risk of the journey (let us call it the risk of the road). If the journey includes more than two stations, the risk of the journey is the sum of risks of the used roads. </p> <p> A direct journey from the North Pole station to the South Pole station has the length π<var>r</var> and risk 1. The journey from the North Pole station to the South Pole station via (0,<var>r</var>,0) has the same length, but a smaller risk: (½π<var>r</var>/(π<var>r</var>))<sup>2</sup>+(½π<var>r</var>/(π<var>r</var>))<sup>2</sup>=0.5. </p> <p> The minimal risk of a journey from the North Pole station to the South Pole station on C(<var>r</var>) is M(<var>r</var>). </p> <p> You are given that M(7)=0.1784943998 rounded to 10 digits behind the decimal point. </p> <p> Find ∑M(2<sup>n</sup>-1) for 1≤n≤15. </p> <p> Give your answer rounded to 10 digits behind the decimal point in the form a.bcdefghijk. </p> </div> <br> <div style="text-align:center;" class="noprint"> Answer: <input type="text" name="answer" id="answer" style="border:1px solid #999;width:100px;"> <button onclick="check_ans(document.getElementById('answer').value);">Check</button> </div> <script type="text/javascript">document.getElementById('answer').focus();</script><div>Go to back to <a href="problems%3Bpage=8.html">Problems</a> </div> </div> <div id="footer" class="noprint">© Project Euler</div> <!--/Creative Commons License--><!-- <rdf:RDF xmlns="http://web.resource.org/cc/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#"> <Work rdf:about=""> <license rdf:resource="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/" /> <dc:type rdf:resource="http://purl.org/dc/dcmitype/Text" /> </Work> <License rdf:about="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/"><permits rdf:resource="http://web.resource.org/cc/Reproduction"/><permits rdf:resource="http://web.resource.org/cc/Distribution"/><requires rdf:resource="http://web.resource.org/cc/Notice"/><requires rdf:resource="http://web.resource.org/cc/Attribution"/><prohibits rdf:resource="http://web.resource.org/cc/CommercialUse"/><permits rdf:resource="http://web.resource.org/cc/DerivativeWorks"/><requires rdf:resource="http://web.resource.org/cc/ShareAlike"/></License></rdf:RDF> --> </div> <div style="height:1px;"> </div> </body> </html>