{"id":3987,"date":"2012-11-30T20:00:57","date_gmt":"2012-11-30T23:00:57","guid":{"rendered":"http:\/\/scienceblogs.com.br\/hypercubic\/?p=3987"},"modified":"2012-11-30T20:00:57","modified_gmt":"2012-11-30T23:00:57","slug":"caf-da-manh-dos-anes","status":"publish","type":"post","link":"https:\/\/www.blogs.unicamp.br\/hypercubic\/2012\/11\/caf-da-manh-dos-anes\/","title":{"rendered":"Caf\u00e9 da manh\u00e3 dos an\u00f5es"},"content":{"rendered":"<p align=\"justify\"><a href=\"http:\/\/scienceblogs.com.br\/hypercubic\/files\/2012\/11\/anes-dO-hobbit.jpg\" data-rel=\"lightbox-image-0\" data-rl_title=\"\" data-rl_caption=\"\" title=\"\"><img fetchpriority=\"high\" decoding=\"async\" style=\"padding-left: 0px;padding-right: 0px;float: none;margin-left: auto;margin-right: auto;padding-top: 0px;border: 0px\" src=\"http:\/\/scienceblogs.com.br\/hypercubic\/files\/2012\/11\/anes-dO-hobbit_thumb.jpg\" alt=\"an\u00f5es d'O hobbit\" width=\"612\" height=\"319\" border=\"0\" \/><\/a><\/p>\n<p align=\"justify\">Muito bem, na segunda-feira observamos os <a href=\"http:\/\/scienceblogs.com.br\/hypercubic\/2012\/11\/enigma-caf-da-manh-dos-anes\/\">sete an\u00f5es na mesa de caf\u00e9 da manh\u00e3<\/a>, partilhando o leite suas respectivas x\u00edcaras. A pergunta era: quanto leite cada x\u00edcara cont\u00e9m, dado que o total de leite era de 420ml?<!--more--><\/p>\n<h3 align=\"justify\">Solu\u00e7\u00e3o dos Leitores<\/h3>\n<p align=\"justify\">Alguns leitores sugeriram que a solu\u00e7\u00e3o consiste em encontrar uma progress\u00e3o aritm\u00e9tica cuja soma seja 420. \u00c9 uma boa estrat\u00e9gia, mas h\u00e1 mais de uma raz\u00e3o poss\u00edvel e apenas uma delas \u00e9 a correta. O <a href=\"http:\/\/scienceblogs.com.br\/hypercubic\/2012\/11\/enigma-caf-da-manh-dos-anes\/comment-page-1\/#comment-13677\">Flavim<\/a> achou que <strong>r = 15ml<\/strong>:<\/p>\n<blockquote><p>a sacada \u00e9 descobrir uma progress\u00e3o aritm\u00e9tica de 7 numeros q some 420<\/p>\n<p>Meus pouquinhos seriam na verdade de 15 ml\u2026<\/p>\n<p>Pelas minhas contas a sequencia seria A1= 105, A2 = 90, A3=75, A4 = 60, A5 = 45, A6 = 30, A7 = 15.<\/p><\/blockquote>\n<p>O <a href=\"http:\/\/scienceblogs.com.br\/hypercubic\/2012\/11\/enigma-caf-da-manh-dos-anes\/comment-page-1\/#comment-13706\">gerson<\/a> tamb\u00e9m:<\/p>\n<blockquote>\n<p align=\"justify\">1\u00ba an\u00e3o com 105ml;<br \/>\n2\u00ba an\u00e3o com 90ml<br \/>\n3\u00ba an\u00e3o com 75ml<br \/>\n4\u00ba an\u00e3o com 60ml<br \/>\n5\u00ba an\u00e3o com 45ml<br \/>\n6\u00ba an\u00e3o com 30ml<br \/>\n7\u00ba an\u00e3o com os restantes 15ml<\/p>\n<\/blockquote>\n<p align=\"justify\">O <a href=\"http:\/\/scienceblogs.com.br\/hypercubic\/2012\/11\/enigma-caf-da-manh-dos-anes\/comment-page-1\/#comment-13485\">Ricardo JJ<\/a> encontrou <strong>r = 20ml<\/strong> e foi mais did\u00e1tico:<\/p>\n<blockquote><p>O \u00faltimo an\u00e3o tem a sua x\u00edcara pura. Isso \u00e9 certo.<\/p>\n<p>Agora os outros, acho que \u00e9 assim:<br \/>\nO sexto an\u00e3o s\u00f3 tem um \u201cpouquinho\u201d de leite que o s\u00e9timo despejou, o quinto tem \u201cdois pouquinhos\u201d e assim por diante at\u00e9 o primeiro an\u00e3o, que vai ter \u201cseis pouquinhos\u201d em sua x\u00edcara. Um total de \u201c21 pouquinhos\u201d, contendo ent\u00e3o 20 ml cada \u201cpouquinho\u201d de leite.<\/p>\n<p>Desse modo, o primeiro an\u00e3o fica com (6*20)=120ml, o segundo com 100ml, o terceiro com 80ml, o quarto com 60ml, o quinto com 40ml e o sexto com 20ml.<\/p><\/blockquote>\n<p style=\"text-align: justify\">Dada a possibilidade de mais de uma raz\u00e3o, ambas as solu\u00e7\u00f5es s\u00e3o poss\u00edveis, mas apenas uma \u00e9 a correta. Quem acertou foi o Ricardo. Parab\u00e9ns!<\/p>\n<h3 align=\"justify\">Solu\u00e7\u00e3o Oficial<\/h3>\n<p align=\"justify\">Este enigma foi adaptado de um problema da Olimp\u00edada Nacional de Matem\u00e1tica da Uni\u00e3o Sovi\u00e9tica de 1977. A solu\u00e7\u00e3o era mais ou menos assim:<\/p>\n<blockquote>\n<p align=\"justify\">Uma vez que o estado final \u00e9 igual ao estado inicial, podemos imaginar que esse caf\u00e9 da manh\u00e3 dos an\u00f5es \u00e9 um processo cont\u00ednuo.<\/p>\n<p align=\"justify\">Considere o an\u00e3o cuja x\u00edcara cont\u00e9m a menor quantidade de leite antes que ele comece a distribui-lo. Digamos que esse an\u00e3o seja o <strong>Fili<\/strong> e\u00a0 que <strong>\u2113<\/strong> \u00e9 uma quantia qualquer de leite (em millitros). Uma vez que a x\u00edcara de <strong>Fili<\/strong> cont\u00e9m a menor quantia de leite antes da divis\u00e3o, cada um dos outros an\u00f5es deve ter uma quantidade de leite <em>maior ou igual<\/em> do que <strong>Fili<\/strong> antes da divis\u00e3o.<\/p>\n<p align=\"justify\">Isso significa que <strong>Fili<\/strong> tem <em>pelo menos<\/em><strong> \u2113\/6<\/strong> millitros, o que deixaria <strong>Fili<\/strong> com pelo menos <strong>\u2113 <\/strong>mililitros em seu pr\u00f3prio copo antes de entorn\u00e1-lo. O \u00fanico modo pelo qual <strong>Fili<\/strong> pode receber precisamente <strong>\u2113<\/strong> mililitros \u2014 e sabemos que \u00e9 isso que ele recebe \u2014 \u00e9 se cada um dos outros seis an\u00f5es lhe passa precisamente <strong>\u2113\/6<\/strong> millitros e nada mais. Isso significa que cada x\u00edcara, logo antes de ser entornada, cont\u00e9m a mesma quantia de leite. Ent\u00e3o, ap\u00f3s qualquer distribui\u00e7\u00e3o as x\u00edcaras ficam com <strong>\u2113<\/strong>, <strong>5\/6\u2113<\/strong>, <strong>4\/6\u2113<\/strong>, <strong>3\/6\u2113<\/strong>, <strong>2\/6\u2113<\/strong>, <strong>1\/6\u2113<\/strong> e zero. Se h\u00e1 um total de <strong>420ml <\/strong>(cortesia de Bilbo Bolseiro), isso resulta em <strong>120<\/strong>, <strong>100<\/strong>, <strong>80<\/strong>, <strong>60<\/strong>, <strong>40<\/strong>, <strong>20<\/strong> e <strong>0<\/strong> mililitros.<\/p>\n<\/blockquote>\n<p align=\"justify\">Evidentemente, o problema original n\u00e3o era ilustrado pelos an\u00f5es de J.R.R. Tolkien.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Muito bem, na segunda-feira observamos os sete an\u00f5es na mesa de caf\u00e9 da manh\u00e3, partilhando o leite suas respectivas x\u00edcaras. A pergunta era: quanto leite cada x\u00edcara cont\u00e9m, dado que o total de leite era de 420ml?<\/p>\n","protected":false},"author":486,"featured_media":4598,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_eb_attr":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"material-hide-sections":[],"pgc_sgb_lightbox_settings":"","_vp_format_video_url":"","_vp_image_focal_point":[],"footnotes":""},"categories":[152],"tags":[],"class_list":["post-3987","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-geral"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.7 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Caf\u00e9 da manh\u00e3 dos an\u00f5es - hypercubic<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.blogs.unicamp.br\/hypercubic\/2012\/11\/caf-da-manh-dos-anes\/\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Renato Pincelli\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/2012\\\/11\\\/caf-da-manh-dos-anes\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/2012\\\/11\\\/caf-da-manh-dos-anes\\\/\"},\"author\":{\"name\":\"Renato Pincelli\",\"@id\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/#\\\/schema\\\/person\\\/c7be1e2811d061a75d3b4f4960eb53bd\"},\"headline\":\"Caf\u00e9 da manh\u00e3 dos an\u00f5es\",\"datePublished\":\"2012-11-30T23:00:57+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/2012\\\/11\\\/caf-da-manh-dos-anes\\\/\"},\"wordCount\":551,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/2012\\\/11\\\/caf-da-manh-dos-anes\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/wp-content\\\/uploads\\\/sites\\\/249\\\/2012\\\/11\\\/anes-dO-hobbit_thumb1.jpg\",\"articleSection\":[\"Geral\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/2012\\\/11\\\/caf-da-manh-dos-anes\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/2012\\\/11\\\/caf-da-manh-dos-anes\\\/\",\"url\":\"https:\\\/\\\/www.blogs.unicamp.br\\\/hypercubic\\\/2012\\\/11\\\/caf-da-manh-dos-anes\\\/\",\"name\":\"Caf\u00e9 da manh\u00e3 dos an\u00f5es - 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