{"id":11076,"date":"2012-10-30T01:11:49","date_gmt":"2012-10-30T01:11:49","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11076"},"modified":"2022-01-20T01:40:13","modified_gmt":"2022-01-20T01:40:13","slug":"hexagonos-e-numeros","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11076","title":{"rendered":"Hex\u00e1gonos e n\u00fameros"},"content":{"rendered":"<p><ul id='GTTabs_ul_11076' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11076' class='GTTabs_curr'><a  id=\"11076_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11076' ><a  id=\"11076_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_11076'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<div id=\"attachment_11077\" style=\"width: 327px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-11077\" data-attachment-id=\"11077\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11077\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros.jpg\" data-orig-size=\"396,384\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Hex\u00e1gonos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros.jpg\" class=\"wp-image-11077\" title=\"Hex\u00e1gonos\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros.jpg\" alt=\"\" width=\"317\" height=\"307\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros.jpg 396w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-300x290.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-150x145.jpg 150w\" sizes=\"auto, (max-width: 317px) 100vw, 317px\" \/><\/a><p id=\"caption-attachment-11077\" class=\"wp-caption-text\">Hex\u00e1gonos e n\u00fameros<\/p><\/div>\n<p>Copia e completa a figura seguinte, inserindo nos c\u00edrculos vazios os n\u00fameros $ + 1$ ou $ &#8211; 1$ de modo que o produto dos seis c\u00edrculos centrados nos v\u00e9rtices de cada um dos hex\u00e1gonos seja sempre igual. Qual \u00e9 esse produto?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11076' onClick='GTTabs_show(1,11076)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11076'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<div id=\"attachment_11080\" style=\"width: 430px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-sol.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-11080\" data-attachment-id=\"11080\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11080\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-sol.jpg\" data-orig-size=\"755,718\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1351559292&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Hex\u00e1gonos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-sol.jpg\" class=\"wp-image-11080\" title=\"Hex\u00e1gonos\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-sol.jpg\" alt=\"\" width=\"420\" height=\"399\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-sol.jpg 755w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-sol-300x285.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-sol-150x142.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/hex\u00e1gononumeros-sol-400x380.jpg 400w\" sizes=\"auto, (max-width: 420px) 100vw, 420px\" \/><\/a><p id=\"caption-attachment-11080\" class=\"wp-caption-text\">SOLU\u00c7\u00c3O<\/p><\/div>\n<p>O produto \u00e9 $ &#8211; 1$.<\/p>\n<p>A solu\u00e7\u00e3o \u00e9 a apresentada ao lado.<\/p>\n<p style=\"text-align: center;\">\n<p><strong>Explica\u00e7\u00e3o<\/strong>:<\/p>\n<p>Comecemos pelo hex\u00e1gono superior esquerdo. H\u00e1 duas possibilidades: preencher o c\u00edrculo vazio com o n\u00famero $ &#8211; 1$ ou com o n\u00famero $ + 1$.<\/p>\n<p>Se escolhermos a primeira possibilidade, obtemos o produto: $ &#8211; 1 \\times 1 \\times \\left( { &#8211; 1} \\right) \\times 1 \\times \\left( { &#8211; 1} \\right) \\times \\left( { &#8211; 1} \\right) = 1$.<\/p>\n<p>Este ser\u00e1 o produto que devemos manter nos restantes hex\u00e1gonos, pelo que devemos ir preenchendo os c\u00edrculos vazios de forma a obter esse valor. No entanto, h\u00e1 uma altura em que n\u00e3o \u00e9 poss\u00edvel obter esse valor para produto.<\/p>\n<p>Desta forma, h\u00e1 que tentar preencher os c\u00edrculos vazios\u00a0partindo da segunda possibilidade, que, como \u00e9 mostrado acima, constitui\u00a0a solu\u00e7\u00e3o do problema.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11076' onClick='GTTabs_show(0,11076)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Copia e completa a figura seguinte, inserindo nos c\u00edrculos vazios os n\u00fameros $ + 1$ ou $ &#8211; 1$ de modo que o produto dos seis c\u00edrculos centrados nos v\u00e9rtices de&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20726,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[317,97,318],"tags":[428,320,319],"series":[],"class_list":["post-11076","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-7-o-ano","category-aplicando","category-numeros-inteiros","tag-7-o-ano","tag-multiplicacao","tag-numeros-inteiros-2"],"views":1627,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/10\/7V1Pag033-1_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11076","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11076"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11076\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20726"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11076"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11076"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11076"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11076"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}