{"id":682,"date":"2021-05-12T12:00:00","date_gmt":"2021-05-12T04:00:00","guid":{"rendered":"https:\/\/blog.cruciferslab.net\/?p=682"},"modified":"2024-03-08T11:20:04","modified_gmt":"2024-03-08T03:20:04","slug":"project-euler-%e8%a7%a3%e9%a1%8c%e7%ad%86%e8%a8%98-4-184","status":"publish","type":"post","link":"https:\/\/blog.cruciferslab.net\/?p=682","title":{"rendered":"Project Euler \u89e3\u984c\u7b46\u8a18 (4) &#8211; #184"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><a href=\"https:\/\/projecteuler.net\/problem=184\">#184 Triangles containing the origin<\/a> (\u96e3\u5ea6 75%)<\/h2>\n\n\n\n<p>\u9019\u984c\u984c\u76ee\u63cf\u8ff0\u8d77\u4f86\u5f88\u7c21\u55ae\uff1a\u8003\u616e\u5e73\u9762\u4e0a\u4e00\u4e9b\u9ede\u7684\u96c6\u5408\uff0c\u9019\u4e9b\u9ede\u6eff\u8db3\uff1a\u9ede\u7684\u5ea7\u6a19\u70ba\u6574\u6578\uff0c\u4ee5\u53ca\u5b83\u5011\u90fd\u5728\u4e00\u500b\u5713\u5fc3\u5728\u539f\u9ede\uff0c\u534a\u5f91\u70ba r \u7684\u5713\u7684\u5167\u90e8\u3002\u8003\u616e\u7531\u9019\u9ede\u96c6\u7576\u4e2d\u7684\u9ede\u70ba\u9802\u9ede\u7684\u4e09\u89d2\u5f62\uff0c\u8a08\u7b97\u5713\u5fc3 (\u539f\u9ede) \u5728\u5176\u5167\u90e8\u7684\u4e09\u89d2\u5f62\u500b\u6578\u3002<br>\u5c0d r=2\uff0c\u4e00\u5171\u53ef\u4ee5\u627e\u5f97\u5230 8 \u500b\u539f\u9ede\u5728\u5167\u90e8\u7684\u4e09\u89d2\u5f62\uff1br=3 \u6709 360 \u500b\uff0c\u800c r=5 \u5247\u6709 10600 \u500b\u3002\u8a66\u6c42 r=105 \u6642\u6709\u591a\u5c11\u500b\u3002<\/p>\n\n\n\n<p>\u770b\u8d77\u4f86\u5f88\u7c21\u55ae\uff0c\u4f46\u4ed4\u7d30\u60f3\u4e0b\u53bb\u5c31\u6703\u767c\u73fe\u5f88\u591a\u7d30\u7bc0\u85cf\u5728\u88e1\u9762\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u9996\u5148\uff0c\u9ede\u7684\u6578\u91cf\u5f88\u591a\uff1a\u7528\u5713\u9762\u7a4d\u4f30\u8a08\u53ef\u4ee5\u5f97\u5230\u534a\u5f91 105 \u7684\u5713\u5167\u6574\u9ede\u6578\u7d04\u6709\u4e09\u842c\u56db\u5343\u591a\u500b\uff0c\u5728\u5176\u4e2d\u9078\u4e09\u500b\u7684\u9078\u6cd5\u6709\u8fd1\u4e03\u5146\u7a2e\uff1b<\/li>\n\n\n\n<li>\u4e09\u89d2\u5f62\u8981\u5305\u542b\u5713\u5fc3\u3002\u8aaa\u8d77\u4f86\u7c21\u55ae\uff0c\u4f46\u5305\u542b\u9019\u56de\u4e8b\u4e26\u4e0d\u5bb9\u6613\u5316\u6210\u689d\u4ef6\uff1b<\/li>\n\n\n\n<li>\u770b\u8d77\u4f86\u597d\u50cf\u6709\u4e00\u4e9b\u5c0d\u7a31\u6027\u53ef\u4ee5\u7528\uff0c\u4f46\u9019\u53c8\u4e0d\u662f\u5927\u5bb6\u90fd\u4e00\u6a23\u7684\u5c0d\u7a31 (\u6709\u7684\u7ffb\u8f49\u4e0d\u540c\uff0c\u6709\u7684\u7ffb\u8f49\u76f8\u540c\u7b49\u7b49)\uff0c\u8981\u5206\u72c0\u6cc1\u6703\u5f88\u9ebb\u7169\uff1b<\/li>\n\n\n\n<li>\u7b49\u7b49\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u5148\u4f86\u63d0\u300c\u5305\u542b\u300d\u9019\u4ef6\u4e8b\u597d\u4e86\uff1b\u56e0\u70ba\u9019\u984c\u5176\u5be6\u662f\u500b\u6a19\u6e96\u7684<a href=\"https:\/\/zh.wikipedia.org\/wiki\/%E8%AE%A1%E7%AE%97%E5%87%A0%E4%BD%95\">\u8a08\u7b97\u5e7e\u4f55<\/a>\u984c\u76ee\u3002\u6240\u8b02\u8a08\u7b97\u5e7e\u4f55\uff0c\u4ee5\u4e0d\u90a3\u9ebc\u7cbe\u6e96\u7684\u61f6\u4eba\u5305\u8aaa\u6cd5\u5c31\u662f\uff1a\u628a\u5e7e\u4f55\u554f\u984c\u5ea7\u6a19\u5316\u7684\u89e3\u6790\u7b97\u5f0f\u7528\u96fb\u8166\u7a0b\u5f0f\u8a08\u7b97\u7684\u6f14\u7b97\u6cd5\u3002\u7562\u7adf\u96fb\u8166\u8981\u300c\u770b\u300d\u4e00\u500b\u5e7e\u4f55\u5716\u5f62\u53ea\u80fd\u7528\u63cf\u8ff0\u5b83\u7684\u53c3\u6578\u6027\u8cea\u53bb\u770b\uff0c\u4e00\u4e9b\u53ea\u8981\u5716\u756b\u51fa\u4f86\u5c0d\u4eba\u4f86\u8aaa\u5f88\u76f4\u63a5\u7684\u5224\u65b7\uff0c\u96fb\u8166\u7a0b\u5f0f\u90fd\u9700\u8981\u9032\u884c\u4e00\u4e9b\u904b\u7b97\u624d\u80fd\u5224\u5b9a\uff1b\u52a0\u4e0a\u4e00\u4e9b\u5e7e\u4f55\u4e0a\u7684\u7c21\u55ae\u6027\u8cea\u5176\u5be6\u5f88\u5bb9\u6613\u51fa\u73fe\u7121\u7406\u6578 (\u4ee5\u6700\u5e38\u898b\u7684\u76f4\u89d2\u4e09\u89d2\u5f62\u4f86\u8aaa\u597d\u4e86\uff0c\u7562\u6c0f\u5b9a\u7406\u6c42\u659c\u908a\u6703\u6709\u6839\u865f\uff0c\u800c\u89d2\u5ea6\u5247\u5c31\u9023\u6709\u7406\u6578\u5ea6\u90fd\u662f\u5c11\u6578)\uff0c\u82e5\u6c92\u6709\u7279\u5225\u8a2d\u8a08\u7684\u8a71\u6578\u503c\u7cbe\u78ba\u5ea6\u4e5f\u6703\u662f\u4e00\u500b\u554f\u984c\u3002<\/p>\n\n\n\n<p>\u5728\u8a08\u7b97\u5e7e\u4f55\u7576\u4e2d\u8981\u6c42<a href=\"https:\/\/en.wikipedia.org\/wiki\/Point_in_polygon\">\u4e00\u500b\u9ede\u662f\u5426\u5305\u5728\u7c21\u55ae\u591a\u908a\u5f62<\/a>\u7576\u4e2d\uff0c\u5e38\u7528\u7684\u505a\u6cd5\u6709\u5e7e\u500b\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7e5e\u591a\u908a\u5f62\u7684\u908a\uff0c\u6bcf\u6b21\u90fd\u770b\u8a72\u9ede\u662f\u5426\u5728\u908a\u7684\u540c\u4e00\u5074\uff1a\u60f3\u50cf\u6cbf\u8457\u591a\u908a\u5f62\u958b\u8eca\uff0c\u7d50\u679c\u7e5e\u4e86\u4e00\u5708\u90fd\u767c\u73fe\u76ee\u6a19\u9ede\u90fd\u5728\u540c\u4e00\u908a\uff0c\u90a3\u5c31\u53ef\u4ee5\u77e5\u9053\u9019\u500b\u9ede\u5728\u591a\u908a\u5f62\u88e1\u9762\u4e86\u3002\u5f9e\u9019\u500b\u5f62\u8c61\u5316\u4e5f\u5f88\u5bb9\u6613\u77e5\u9053\u77e5\u9053\u9019\u500b\u65b9\u6cd5\u53ea\u5c0d\u51f8\u591a\u908a\u5f62\u6709\u7528\uff0c\u4e0d\u904e\u4e09\u89d2\u5f62\u4e00\u5b9a\u662f\u51f8\u7684\u6240\u4ee5\u5c0d\u4e09\u89d2\u5f62\u4e5f\u7b97\u5e38\u7528\u3002<\/li>\n\n\n\n<li>\u5f9e\u9ede\u4efb\u610f\u767c\u5c04\u5c04\u7dda\uff0c\u5982\u679c\u53ea\u548c\u591a\u908a\u5f62\u4ea4\u53c9\u5947\u6578\u6b21\u90a3\u5c31\u662f\u5728\u88e1\u9762\uff1a\u9019\u4e5f\u662f\u500b\u5728\u5716\u5f62\u4e0a\u6eff\u76f4\u89c0\u7684\u505a\u6cd5\u3002<\/li>\n\n\n\n<li>\u4e0a\u4e00\u500b\u65b9\u6cd5\u7684\u9032\u968e\u7248\uff1a\u8a08\u7b97\u9019\u500b\u591a\u908a\u5f62\u5c0d\u6240\u6c42\u9ede\u7684<a href=\"https:\/\/zh.wikipedia.org\/wiki\/%E5%8D%B7%E7%BB%95%E6%95%B0\">\u5377\u7e5e\u6578<\/a>\uff0c\u5982\u679c\u4e0d\u662f\u96f6\u90a3\u5c31\u662f\u5728\u88e1\u9762\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u7b49\u7b49\u3002<\/p>\n\n\n\n<p>\u55ae\u55ae\u53ea\u904b\u7528\u9019\u4e9b\u65b9\u5f0f\u5c31\u80fd\u8f15\u9b06\u5beb\u7a0b\u5f0f\u6aa2\u9a57\u984c\u76ee\u7d66\u7684\u4e09\u500b\u6578\u5b57\uff0c\u4f46\u76f4\u4e0a r=105 \u6642\u5c31\u6703\u5361\u4f4f\u4e86\u3002\u9019\u624d\u662f\u9019\u984c\u7684\u56f0\u96e3\u9ede\uff1a\u9ede\u96c6\u5167\u6709 \\(O(r^2)\\) \u500b\u9ede\uff0c\u6240\u4ee5\u9078\u4e09\u500b\u9ede\u7d44\u6210\u7684\u4e09\u89d2\u5f62\u500b\u6578\u6709 \\(O(r^6)\\) \u500b\uff0c\u5168\u90e8\u8a66\u904e\u4e00\u904d\u7684\u6642\u9593\u6703\u592a\u4e45\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/projecteuler.net\/profile\/progheal.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<!--more-->\n\n\n\n<p>\u4e0a\u9762\u7b2c\u4e09\u500b\u65b9\u6cd5\u63d0\u5230\u7684\u5377\u7e5e\u6578\uff0c\u4ed4\u7d30\u601d\u8003\u53ef\u4ee5\u767c\u73fe\u5c0d\u65bc\u591a\u908a\u5f62\u5176\u5be6\u53ef\u4ee5\u9032\u884c\u7c21\u5316\uff1a\u56e0\u70ba\u591a\u908a\u5f62\u7684\u908a\u662f\u76f4\u7dda\uff0c\u6cbf\u8457\u908a\u7e5e\u6642\u6211\u5011\u53ef\u4ee5\u7531\u8d77\u9ede\u548c\u7d42\u9ede\u7684\u65b9\u5411\u77e5\u9053\u9019\u908a\u6703\u589e\u52a0\u6216\u6e1b\u5c11\u5377\u7e5e\u6578\uff1b\u800c\u82e5\u5305\u9019\u4e00\u9ede\u7684\u662f\u7c21\u55ae\u591a\u908a\u5f62\uff0c\u5377\u7e5e\u6578\u53ea\u6703\u662f 1 \u6216 -1 (\u770b\u662f\u7e5e\u54ea\u500b\u65b9\u5411)\uff0c\u56e0\u6b64\u5c0d\u65bc\u4e09\u89d2\u5f62\u5c31\u53ef\u4ee5\u7c21\u5316\u6210\uff1a\u6bcf\u500b\u908a\u7e5e\u7684\u65b9\u5411\u90fd\u4e00\u6a23\u7684\u8a71\u5b83\u5c31\u6703\u5305\u4f4f\u539f\u9ede\u3002<\/p>\n\n\n\n<p>\u65e2\u7136\u6211\u5011\u53ea\u8981\u5168\u90e8\u76f8\u540c\u7684\u65b9\u5411\u7684\u8a71\uff0c\u90a3\u4e0d\u5982\u6211\u5011\u81ea\u5df1\u53d6\uff0c\u9019\u6a23\u9084\u53ef\u4ee5\u7701\u53bb\u5224\u65b7\uff1b\u53ea\u8981\u627e\u4e00\u500b\u53d6\u6cd5\u80fd\u5920\u628a\u6240\u6709\u8a72\u7b97\u5f97\u5230\u7684\u4e09\u89d2\u5f62\u90fd\u7b97\u5230\u5c31\u884c\u4e86\u3002\u65e2\u7136\u4e0a\u9762\u7684\u7c21\u5316\u7248\u662f\u7531\u9802\u9ede\u5230\u9802\u9ede\u9019\u6a23\u6383\uff0c\u90a3\u6211\u5011\u53ef\u4ee5\u5148\u628a\u6240\u6709\u53ef\u80fd\u7684\u9802\u9ede\u6240\u5728\u65b9\u5411\u7b97\u51fa\u4f86\uff0c\u90a3\u7531\u4e00\u500b\u9ede\u6383\u5230\u53e6\u4e00\u500b\u9ede\u5c31\u53ea\u8981\u770b\u9019\u5169\u500b\u9ede\u7684\u65b9\u5411\u5c31\u80fd\u5224\u65b7\u9019\u4e00\u6383\u662f\u6b63\u5411\u9084\u662f\u53cd\u5411\u3002\u9019\u6a23\u4e00\u4f86\uff0c\u96a8\u4fbf\u53d6\u5169\u500b\u9ede\uff0c\u7b2c\u4e09\u500b\u9ede\u7684\u6240\u5728\u65b9\u5411\u662f\u6709\u9650\u5b9a\u7684\uff1a\u4f8b\u5982\u4e0b\u5716\uff0c\u53ea\u6709\u5728\u8457\u8272\u5340\u57df\u7576\u4e2d\u7684 C \u9ede\u624d\u80fd\u548c A B \u5408\u6210\u5305\u542b O \u9ede\u7684\u4e09\u89d2\u5f62\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"731\" height=\"723\" src=\"https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/05\/image.png\" alt=\"\" class=\"wp-image-685\" style=\"width:525px;height:520px\" srcset=\"https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/05\/image.png 731w, https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/05\/image-300x297.png 300w, https:\/\/blog.cruciferslab.net\/wp-content\/uploads\/2021\/05\/image-150x148.png 150w\" sizes=\"auto, (max-width: 731px) 100vw, 731px\" \/><\/figure>\n<\/div>\n\n\n<p>\u601d\u8003\u5230\u9019\u908a\uff0c\u6211\u5011\u7684\u6f14\u7b97\u6cd5\u5c31\u5f88\u6e05\u695a\u4e86\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u679a\u8209\u6240\u6709\u7684 A \u548c B \u7684\u7d44\u5408\u3002A \u53ef\u4ee5\u53ea\u9650\u5b9a\u5728\u4e00\u500b\u534a\u5713\uff0c\u56e0\u70ba\u8981\u5305\u542b\u539f\u9ede\u4efb\u4f55\u4e00\u500b\u534a\u5713\u81f3\u5c11\u6703\u6709\u4e00\u500b\u9ede\uff1b<\/li>\n\n\n\n<li>\u5c0d\u6bcf\u500b\u7d44\u5408\uff0c\u4f7f\u7528\u4e0a\u9762\u7684\u65b9\u5f0f\u627e\u51fa\u53ef\u4ee5\u7528\u7684 C \u9ede\u7bc4\u570d\uff0c\u628a\u9019\u7bc4\u570d\u5167\u7684 C \u9ede\u500b\u6578\u52a0\u9032\u8a08\u7b97\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u56e0\u70ba C \u9ede\u7bc4\u570d\u53ef\u4ee5\u8b93 A \u548c B \u6c7a\u5b9a\uff0c\u6240\u4ee5\u7b2c\u4e00\u500b\u679a\u8209\u7684\u9805\u6578\u6e1b\u5c11\u5230 \\(O(r^4)\\) \u4e86\u3002\u70ba\u4e86\u8981\u80fd\u5feb\u901f\u6c42\u51fa\u4e0a\u9762\u9019\u6a23\u7684\u7bc4\u570d\uff0c\u6211\u5011\u53ef\u4ee5\u628a\u6240\u6709\u9ede\u4f9d\u7167\u65b9\u5411\u6392\u5e8f\uff0c\u90a3\u53ea\u8981 A B \u53d6\u5b9a\u4e86\u4e4b\u5f8c\u5f88\u5bb9\u6613\u5c31\u80fd\u627e\u5230\u5c0d\u9762\u65b9\u5411\u5728\u54ea\u88e1\uff0c\u5c31\u80fd\u7c21\u55ae\u7684\u6c42\u51fa\u7bc4\u570d\u5167\u9ede\u6578\uff0c\u52a0\u7e3d\u8d77\u4f86\u5c31\u662f\u6240\u9700\u8981\u7684\u7b54\u6848\u4e86\u3002<\/p>\n\n\n\n<p>\u2026\u2026\u53ea\u8981\u4f60\u7684\u7a0b\u5f0f\u6c92\u6709\u56e0\u70ba\u6578\u503c\u7cbe\u78ba\u5ea6\u800c\u6f0f\u7b97\u591a\u7b97\u7684\u8a71\u3002<\/p>\n\n\n\n<p>\u662f\u7684\uff0c\u9019\u984c\u7684\u96e3\u95dc\u5176\u5be6\u4e0d\u662f\u6f14\u7b97\u6cd5\uff0c\u800c\u662f\u7576\u4f60\u76f4\u63a5\u7528\u6578\u5b78\u95dc\u4fc2\u5beb\u7a0b\u5f0f\u6642\u6703\u9047\u4e0a\u7684\u6578\u503c\u7cbe\u78ba\u5ea6\u554f\u984c\u3002<\/p>\n\n\n\n<p>\u554f\u984c\u662f\u5728\u5c07\u9ede\u4f9d\u65b9\u5411\u6392\u5e8f\u4e0a\u3002\u6392\u5e8f\u4e0d\u662f\u554f\u984c\uff0c\u4f46\u9019\u500b\u300c\u65b9\u5411\u300d\uff0c\u6709\u4e00\u9ede\u4e09\u89d2\u51fd\u6578\u5e95\u5b50\u7684\u4eba\u61c9\u8a72\u6703\u76f4\u89ba\u60f3\u5230 <code>atan<\/code> (\u53cd\u6b63\u5207) \u51fd\u6578\u5f9e\u659c\u7387\u6c42\u89d2\u5ea6\uff0c\u6bd4\u8f03\u6709\u6ce8\u610f\u5230\u7d30\u7bc0\u7684\u4eba\u53ef\u80fd\u9084\u6703\u4f7f\u7528\u6703\u8003\u616e\u9032\u8c61\u9650\u7684 <code>atan2<\/code> \u4f86\u6c42\u65b9\u5411\uff1b\u554f\u984c\u5728\u65bc\uff0c\u56e0\u70ba\u6211\u5011\u6709\u7684\u662f\u6574\u6578\u9ede\uff0c\u53cd\u6b63\u5207\u51fd\u6578\u5e7e\u4e4e\u4e0d\u6703\u51fa\u73fe\u6709\u7406\u6578\u7d50\u679c (\u6574\u6578\u9ede\u53ea\u6709\u6b63 x \u65b9\u5411\u7684\u9ede\u7684\u53cd\u6b63\u5207\u503c 0 \u662f\u6709\u7406\u6578\u800c\u5df2)\uff0c\u9019\u6a23\u4e00\u4f86\u540c\u4e00\u500b\u65b9\u5411\u7684\u7d50\u679c\u6709\u53ef\u80fd\u6703\u6709\u4e0d\u540c\u7cbe\u78ba\u5ea6\u7684\u7b54\u6848\u3002\u56e0\u70ba\u9019\u500b\u95dc\u4fc2\uff0c\u8981\u600e\u9ebc\u6c7a\u5b9a\u4e00\u500b\u65b9\u5411\u7684\u5c0d\u9762\u5f9e\u54ea\u88e1\u958b\u59cb\u4e5f\u6703\u662f\u500b\u554f\u984c\uff1a180 \u5ea6\u662f \\(\\pi\\) \u5f27\uff0c\u800c \\(\\pi\\) \u662f\u7121\u7406\u6578\uff0c\u5df2\u77e5\u65b9\u5411\u7684\u89d2\u4e5f\u662f\u7121\u7406\u6578\uff0c\u8981\u78ba\u5b9a\u52a0 \\(\\pi\\) \u7684\u89d2\u5ea6\u9644\u8fd1\u54ea\u4e9b\u4e5f\u662f\u4e00\u6a23\u7684\u89d2\u5ea6\u6709\u9ede\u9ebb\u7169\u3002\u9019\u5c31\u662f\u9020\u6210\u4e0a\u9762\u63d0\u5230\u7684\u53ef\u80fd\u6703\u767c\u751f\u6f0f\u7b97\u591a\u7b97\u7684\u539f\u56e0\u3002<\/p>\n\n\n\n<p>\u9019\u500b\u7cbe\u78ba\u5ea6\u5dee\u8ddd\u5176\u5be6\u662f\u53ef\u4ee5\u7d66\u51fa\u4e00\u500b\u4f30\u8a08\u503c\u7684\uff1a\u7531\u6b63\u5207\u5dee\u89d2\u516c\u5f0f\uff0c\u4efb\u5169\u500b\u9ede\u7684\u89d2\u5ea6\u5dee\u70ba\uff1a<br>$$\\tan\\left(\\arctan\\left(\\frac{p}{q}\\right)-\\arctan\\left(\\frac{r}{s}\\right)\\right)=\\frac{ps-qr}{pr+qs}$$<br>\u7531\u65bc\u6574\u9ede\u5ea7\u6a19 \\(p, q, r, s\\) \u4e0d\u8d85\u904e 105\uff0c\u9019\u500b\u503c\u7684\u5206\u5b50\u7d55\u5c0d\u503c\u6700\u5c0f\u662f 1\uff0c\u5206\u6bcd\u5247\u4e0d\u8d85\u904e \\(2\\times105^2=22050\\)\uff0c\u56e0\u6b64\u9019\u500b\u6b63\u5207\u503c\u82e5\u662f\u975e\u96f6\u5247\u4e0d\u5c0f\u65bc \\(\\frac1{22050}\\approx4.535\\times10^{-5}\\)\uff0c\u7531\u6b63\u5207\u7684\u5c0f\u89d2\u5ea6\u8fd1\u4f3c\u4e5f\u5c31\u77e5\u9053\u89d2\u5ea6\u503c\u7684\u5dee\u4e5f\u5927\u7d04\u81f3\u5c11\u6709\u5dee\u9019\u9ebc\u591a\u3002\u6240\u4ee5\u5176\u5be6\u6709\u500b\u6bd4\u8f03\u61f6\u4eba\u7684\u89e3\u6cd5\u5c31\u662f\u7528\u8001\u62db\u8853\uff1a\u53d6\u500b\u6bd4\u9019\u500b\u5c0f\u8a31\u591a\u7684\u8aa4\u5dee\u503c\u7576\u76f8\u7b49\u3002\u4f46\u4e00\u4f86\u5230\u5e95\u8981\u53d6\u591a\u5c11\u624d\u5c0d\u53ea\u80fd\u7528\u731c\u7684\uff0c\u4e8c\u4f86\u9019\u7121\u52a9\u65bc\u7c21\u5316\u300c\u627e\u5c0d\u9762\u300d\u9019\u56de\u4e8b\uff0c\u56e0\u70ba\u4ecd\u7136\u9700\u8981\u6aa2\u67e5\u627e\u5230\u7684\u5c0d\u9762\u65c1\u908a\u6709\u6c92\u6709\u540c\u65b9\u5411\u7684\u9ede\u3002<\/p>\n\n\n\n<p>\u4e0d\u904e\uff0c\u81f3\u5c11\u5728\u9019\u500b\u554f\u984c\u4e0a\uff0c\u6211\u5011\u662f\u6709\u4e00\u500b\u5b8c\u5168\u6c92\u6709\u6d6e\u9ede\u6578\u8aa4\u5dee\u7684\u505a\u6cd5\u7684\uff1a\u76f4\u63a5\u6bd4\u5c0d\u5169\u500b\u6574\u9ede\u7684\u65b9\u5411\u4f4d\u7f6e\u3002\u540c\u6a23\u5229\u7528\u4e0a\u9762\u90a3\u689d\u516c\u5f0f\uff0c\u53ea\u8981\u5169\u500b\u65b9\u5411\u4e0d\u5dee\u8d85\u904e 180 \u5ea6\uff0c\u90a3\u6211\u5011\u5c31\u80fd\u7528\u6b63\u5207\u503c\u5206\u5b50 \\(ps-qr\\) \u7684\u6b63\u8ca0 (\u6216\u8005\u7b49\u50f9\u5730\uff0c\u5176\u6b63\u5f26\u503c\u7684\u6b63\u8ca0) \u4f86\u5224\u65b7\u5169\u500b\u65b9\u5411\u8ab0\u5148\u8ab0\u5f8c\u3002\u4e0d\u5dee\u8d85\u904e 180 \u5ea6\u7684\u5247\u53ef\u4ee5\u7c21\u55ae\u7684\u6c7a\u5b9a\u4e00\u500b\u5c0d\u5207 (\u4f8b\u5982\u6cbf x \u8ef8\u5207\u958b)\uff0c\u4e0d\u540c\u908a\u7684\u76f4\u63a5\u7d66\u7b54\u6848\uff0c\u540c\u4e00\u908a\u7684\u518d\u4f86\u4ee3\u9019\u500b\u516c\u5f0f\u3002\u9019\u6a23\u5c31\u80fd\u4fdd\u6301\u65b9\u5411\u7684\u6574\u9ede\u5ea7\u6a19\u800c\u5c07\u5b83\u5011\u6392\u5e8f\u4e86\u3002<\/p>\n\n\n\n<p>\u4fdd\u6301\u65b9\u5411\u7684\u6574\u9ede\u5ea7\u6a19\u9084\u6709\u4e00\u500b\u597d\u8655\u662f\uff1a\u6211\u5011\u53ef\u4ee5\u52d5\u9ede\u624b\u8173\u8b93\u627e\u5c0d\u9762\u9019\u56de\u4e8b\u662f \\(O(1)\\)\u3002\u6709\u756b\u51fa\u5716\u7684\u4eba\u90fd\u6703\u6ce8\u610f\u5230\uff0c\u540c\u4e00\u500b\u65b9\u5411\u4e0a\u7684\u6574\u9ede\u7e3d\u662f\u6703\u4e00\u8d77\u53c3\u8207\u8a08\u6578\uff1a\u5982\u679c\u6709\u500b\u4e09\u89d2\u5f62\u6709\u5305\u4f4f\u5713\u5fc3\uff0c\u90a3\u628a\u9802\u9ede\u5728\u540c\u4e00\u500b\u65b9\u5411\u4e0a\u79fb\u52d5\u4e5f\u4ecd\u7136\u6703\u5305\u5f97\u5230\u3002\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u505a\u4e00\u9ede\u5c0f\u6574\u7406\uff0c\u628a\u6240\u6709\u540c\u65b9\u5411\u7684\u9ede\u7d66\u96c6\u5408\u5728\u4e00\u8d77\uff0c\u679a\u8209 A B \u6642\u5c31\u4e00\u53e3\u6c23\u679a\u8209\u9019\u500b\u65b9\u5411\u4e0a\u7684\u6240\u6709\u9ede\u4e00\u8d77\u8a08\u7b97\u3002\u9019\u6a23\u4e00\u4f86\uff0c\u300c\u5c0d\u9762\u300d\u9019\u500b\u76ee\u6a19\u8b8a\u55ae\u4e00\u4e86\uff0c\u800c\u5982\u679c\u80fd\u5920\u5229\u7528\u4e00\u4e9b\u8cc7\u6599\u7d50\u69cb\uff0c\u6211\u5011\u751a\u81f3\u53ef\u4ee5\u5728\u6574\u7406\u7684\u6642\u5019\u5c31\u628a\u300c\u5c0d\u9762\u300d\u95dc\u4fc2\u7d66\u5efa\u7acb\u8d77\u4f86<sup data-fn=\"4d935f89-9ea2-40b9-9508-d62b2f570ad2\" class=\"fn\"><a href=\"#4d935f89-9ea2-40b9-9508-d62b2f570ad2\" id=\"4d935f89-9ea2-40b9-9508-d62b2f570ad2-link\">1<\/a><\/sup>\uff0c\u4e4b\u5f8c\u8981\u627e\u7684\u6642\u5019\u76f4\u63a5\u6cbf\u8457\u95dc\u4fc2\u627e\u904e\u53bb\u5c31\u884c\u4e86\u3002\u9019\u500b\u64cd\u4f5c\u4e26\u4e0d\u6703\u6e1b\u5c11\u6642\u9593\u8907\u96dc\u5ea6\uff0c\u56e0\u70ba\u6240\u6392\u5e8f\u7684\u9ede\u53ea\u662f\u5f9e\u6240\u6709\u6578\u5c0d\u8b8a\u6210\u6240\u6709\u4e92\u8cea\u6578\u5c0d\uff0c\u800c\u5927\u7bc4\u570d\u5167\u4efb\u53d6\u5169\u6574\u6578\u4e92\u8cea\u7684\u6a5f\u7387\u662f \\(\\frac6{\\pi^2}\\approx60.8\\%\\) \u662f\u500b\u5e38\u6578\uff0c\u4e5f\u5c31\u662f\u6211\u5011\u53ea\u662f\u628a\u6392\u5e8f\u7684\u9ede\u6578\u6e1b\u5c11\u70ba 60% \u800c\u5df2\uff0c\u4ecd\u7136\u6709 \\(O(r^2)\\) \u500b\u7d00\u9304\uff0c\u56e0\u6b64\u9019\u90e8\u4efd\u7684\u6642\u9593\u8907\u96dc\u5ea6\u662f\u4e0d\u6703\u6539\u8b8a\u7684\uff0c\u4f46\u591a\u4e86\u8655\u7406\u4e0a\u7684\u65b9\u4fbf\u6027\u3002<\/p>\n\n\n\n<p>\u90a3\u9ebc\u627e\u51fa\u7bc4\u570d\u5167\u7684\u9ede\u6578\u5728\u6709\u4e86\u9019\u500b\u6574\u7406\u4e4b\u5f8c\uff0c\u53ea\u8981\u518d\u82b1\u8cbb \\(O(r^2)\\) \u6383\u4e00\u6b21\u505a\u90e8\u4efd\u548c\uff0c\u4e4b\u5f8c\u6bcf\u6b21\u679a\u8209 A B \u6642 \\(O(1)\\) \u627e\u51fa\u5c0d\u9762\u908a\u754c\uff0c\u90e8\u4efd\u548c\u76f8\u6e1b\u5373\u53ef\u6c42\u51fa C \u9ede\u500b\u6578\uff0c\u4e58\u8d77\u4f86\u52a0\u5230\u7e3d\u548c\u5373\u53ef\u3002\u7e3d\u8a08\u6642\u9593\u5c31\u662f\u679a\u8209\u7684 \\(O(r^4)\\)\uff0c\u6578\u79d2\u5167\u5c31\u80fd\u8dd1\u51fa\u7b54\u6848\u4f86\u3002\u5be6\u969b\u7a0b\u5f0f\u7684\u7d30\u7bc0\u5728\u9019\u88e1\u6c92\u6709\u63d0\u7684\u9084\u6709\u5f88\u591a\uff0c\u5c31\u4e0d\u4e00\u4e00\u63d0\u4e86\uff0c\u9806\u4fbf\u7576\u505a\u4e00\u500b\u6709\u9ede\u6545\u610f\u7684\u9632\u6284 XD (\u9019\u4e9b\u7d30\u7bc0\u53ea\u8981\u6709\u4e86\u4e0a\u9762\u7684\u5927\u6982\u5ff5\uff0c\u5728\u5be6\u4f5c\u4e2d\u90fd\u6703\u6ce8\u610f\u5230\u7684)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u8a3b\u8173<\/h2>\n\n\n<ol class=\"wp-block-footnotes\"><li id=\"4d935f89-9ea2-40b9-9508-d62b2f570ad2\">\u9019\u4e00\u9ede\u7531\u65bc\u6bd4\u8f03\u7d30\u7bc0\u5c31\u4e0d\u8a73\u63d0\u4e86\uff0c\u5927\u7565\u662f\u5229\u7528\u4e86 <code>std::map<\/code> \u7684\u5be6\u4f5c\uff0c\u5728\u9032\u884c\u65b9\u5411\u7d71\u8a08\u6642\u540c\u6642\u7372\u5f97\u5c0d\u9762\u65b9\u5411\u7684\u8cc7\u6599\u6240\u5728\uff0c\u800c\u4e14\u9019\u500b\u6240\u5728\u4e0d\u6703\u56e0\u70ba\u5f8c\u7e8c\u65b0\u589e\u8cc7\u6599\u800c\u8b8a\u52d5\u3002 <a href=\"#4d935f89-9ea2-40b9-9508-d62b2f570ad2-link\" aria-label=\"\u8fd4\u56de\u8a3b\u8173\u53c3\u7167 1\">\u21a9\ufe0e<\/a><\/li><\/ol>","protected":false},"excerpt":{"rendered":"<p>#184 Triangles containing the origin (\u96e3\u5ea6 75%) \u9019\u984c\u984c\u76ee\u63cf\u8ff0\u8d77\u4f86\u5f88 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"[{\"content\":\"\u9019\u4e00\u9ede\u7531\u65bc\u6bd4\u8f03\u7d30\u7bc0\u5c31\u4e0d\u8a73\u63d0\u4e86\uff0c\u5927\u7565\u662f\u5229\u7528\u4e86 <code>std::map<\/code> \u7684\u5be6\u4f5c\uff0c\u5728\u9032\u884c\u65b9\u5411\u7d71\u8a08\u6642\u540c\u6642\u7372\u5f97\u5c0d\u9762\u65b9\u5411\u7684\u8cc7\u6599\u6240\u5728\uff0c\u800c\u4e14\u9019\u500b\u6240\u5728\u4e0d\u6703\u56e0\u70ba\u5f8c\u7e8c\u65b0\u589e\u8cc7\u6599\u800c\u8b8a\u52d5\u3002\",\"id\":\"4d935f89-9ea2-40b9-9508-d62b2f570ad2\"}]"},"categories":[8,3,4],"tags":[],"class_list":["post-682","post","type-post","status-publish","format-standard","hentry","category-projecteuler","category-math","category-programming"],"_links":{"self":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/682","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=682"}],"version-history":[{"count":17,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/682\/revisions"}],"predecessor-version":[{"id":1755,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=\/wp\/v2\/posts\/682\/revisions\/1755"}],"wp:attachment":[{"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=682"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=682"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.cruciferslab.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=682"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}