{"id":744,"date":"2009-11-26T21:59:00","date_gmt":"2009-11-26T21:59:00","guid":{"rendered":"http:\/\/wp.andreas.bieri.name\/myblog\/?p=744"},"modified":"2009-11-26T21:59:00","modified_gmt":"2009-11-26T21:59:00","slug":"lsung-quiz-qua-20074","status":"publish","type":"post","link":"http:\/\/52.29.166.97\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/","title":{"rendered":"L\u00f6sung Quiz QUA 2007\/4"},"content":{"rendered":"<p>Wir nennen das Gef\u00e4ss mit 8 Liter Inhalt <b>A<\/b>, das mit 5 heisse <b>B<\/b> und das dritte mit 3 Liter sei <b>C<\/b>.\u00a0Den F\u00fcllstand k\u00f6nnen mir vereinfacht mit einer 3-stelligen Zahl kodieren, z.B. bedeute\u00a0800 = 8 Liter im Gef\u00e4ss <b>A<\/b>, 0 Liter in <b>B<\/b> und 0 in <b>C<\/b>.<\/p>\n<p>Die L\u00f6sung lautet in dieser Notation: 800 \u2192 350 \u2192 323 \u2192 620 \u2192 602 \u2192 152 \u2192 143.<\/p>\n<p>Man kann solche R\u00e4tsel systematisch l\u00f6sen, wenn man bemerkt, dass die Summe (die gesamte Fl\u00fcssigkeitsmenge) aller Zahlen konstant ist. Man kann einen F\u00fcllstand <i>abc<\/i> mit der Bedingung <i>a+b+c = 8<\/i> als Punkt eines Dreiecks auffassen &#8211; vielleicht erinnert man sich noch an die <i>Schwerpunktskoordinaten<\/i> (baryzentrische Koordinaten) aus der Physik?\u00a0Im Dreieck ist 800 oben (Ecke A), 080 links unten (Ecke B), 008 rechts unten (Ecke C). Jeder F\u00fcllstand entspricht einem Punkt im Dreieck.<\/p>\n<p><img loading=\"lazy\" class=\" wp-image-749 aligncenter\" src=\"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-1.jpg\" alt=\"q07-1\" width=\"396\" height=\"378\" srcset=\"http:\/\/52.29.166.97\/myblog\/wp-content\/uploads\/2009\/11\/q07-1.jpg 639w, http:\/\/52.29.166.97\/myblog\/wp-content\/uploads\/2009\/11\/q07-1-300x287.jpg 300w\" sizes=\"(max-width: 396px) 100vw, 396px\" \/><\/p>\n<p>Durch Umsch\u00fctten bewegt man sich in diesem Dreieck nach folgenden Regeln:<\/p>\n<ul>\n<li>Start ist der Punkt 800 oben (alle 8 Liter im Gef\u00e4ss A)<\/li>\n<li>man bleibt auf dem Gitterraster, da die Mengen immer ganze Liter und keine Bruchteile von Litern sind<\/li>\n<li>man bleibt im umrandeten Parallelogramm, die Eckpunkte sind 800, 350, 503, 053. Dann sind 1 oder 2 Gef\u00e4sse ganz leer oder voll. Punkte ausserhalb k\u00f6nnen nicht erreicht werden.<\/li>\n<li>Immer wenn man ein Gef\u00e4ss ganz leert oder f\u00fcllt, &#8222;l\u00e4uft&#8220; man auf den Rand zu und landet schliesslich auf dem Rand.<\/li>\n<li>vielleicht am schwierigsten einzusehen ist die letzte Regel: wenn man wieder vom Rand &#8222;wegl\u00e4uft&#8220;, indem das Wasser umgiesst, gilt <i>Einfallswinkel = Ausfallswinkel<\/i>, d.h. man bewegt sich wie eine Billardkugel!<br \/>\n<img loading=\"lazy\" class=\" wp-image-748 aligncenter\" src=\"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-2.jpg\" alt=\"q07-2\" width=\"351\" height=\"158\" \/><\/li>\n<\/ul>\n<p>Damit stehen wir einfach vor der Aufgabe, durch Billardspielen im Parallelogramm einen Punkt <i>auf dem Rand<\/i> mit 4 Litern zu treffen! M\u00f6gliche Punkte sind nur 440, 413, 143.<img loading=\"lazy\" class=\" wp-image-747 aligncenter\" src=\"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-3.jpg\" alt=\"q07-3\" width=\"422\" height=\"364\" srcset=\"http:\/\/52.29.166.97\/myblog\/wp-content\/uploads\/2009\/11\/q07-3.jpg 671w, http:\/\/52.29.166.97\/myblog\/wp-content\/uploads\/2009\/11\/q07-3-300x259.jpg 300w\" sizes=\"(max-width: 422px) 100vw, 422px\" \/><\/p>\n<p>Zur Zusatzfrage: K\u00f6nnen wir mit 3 Gef\u00e4ssen mit 8, 7, 6 Liter Inhalt auch 5 Liter abmessen? Dies ist nicht l\u00f6sbar, man versuche es! Grafisch etwas einfacher ist die Aufgabe, mit 8,7,6 Liter und <b>10<\/b> Liter Wasser 5 abzumessen: mit den obigen \u00dcberlegungen stellt man fest, dass alle L\u00f6sungen auf einem geschlossenen gleichseitigen Dreieck mit den Ecken 550, 055, 505 liegen. In dieses Dreieck kommt man nur hinein, wenn man schon darauf startet (Billardbedingung!).<br \/>\n<img loading=\"lazy\" class=\"size-full wp-image-746 aligncenter\" src=\"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-4.jpg\" alt=\"q07-4\" width=\"458\" height=\"371\" srcset=\"http:\/\/52.29.166.97\/myblog\/wp-content\/uploads\/2009\/11\/q07-4.jpg 458w, http:\/\/52.29.166.97\/myblog\/wp-content\/uploads\/2009\/11\/q07-4-300x243.jpg 300w\" sizes=\"(max-width: 458px) 100vw, 458px\" \/><\/p>\n<p>Link: <a href=\"http:\/\/www.cut-the-knot.org\/ctk\/Water.shtml\">http:\/\/www.cut-the-knot.org\/ctk\/Water.shtml <\/a>(mit Java-Demo)<\/p>\n<h3><\/h3>\n<p>&nbsp;<\/p>\n<h3><b>Neue kleine Quizfrage<\/b><\/h3>\n<p>Wie kann man die Zahlen 23 und 27 mit der Zahl <b>1<\/b>, Additionen (<b>+<\/b>), Multiplikationen (<b>x<\/b>) und Klammern m\u00f6glichst kurz schreiben? Wie oft wird die Zahl 1 gebraucht ?<\/p>\n<p>Beispiele:<\/p>\n<p>5 = (1+1)\u00d7(1+1)+1<\/p>\n<p>13 = (1+1+1)\u00d7(1+1)\u00d7(1+1)+1<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Wir nennen das Gef\u00e4ss mit 8 Liter Inhalt A, das mit 5 heisse B und das dritte mit 3 Liter sei C.\u00a0Den F\u00fcllstand k\u00f6nnen mir vereinfacht mit einer 3-stelligen Zahl kodieren, z.B. bedeute\u00a0800 = 8 Liter im Gef\u00e4ss A, 0 Liter in B und 0 in C. Die L\u00f6sung lautet in dieser Notation: 800 \u2192 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4],"tags":[105],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v18.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>L\u00f6sung Quiz QUA 2007\/4 - Merkbar.<\/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:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/\" \/>\n<meta property=\"og:locale\" content=\"de_DE\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"L\u00f6sung Quiz QUA 2007\/4 - Merkbar.\" \/>\n<meta property=\"og:description\" content=\"Wir nennen das Gef\u00e4ss mit 8 Liter Inhalt A, das mit 5 heisse B und das dritte mit 3 Liter sei C.\u00a0Den F\u00fcllstand k\u00f6nnen mir vereinfacht mit einer 3-stelligen Zahl kodieren, z.B. bedeute\u00a0800 = 8 Liter im Gef\u00e4ss A, 0 Liter in B und 0 in C. Die L\u00f6sung lautet in dieser Notation: 800 \u2192 [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/\" \/>\n<meta property=\"og:site_name\" content=\"Merkbar.\" \/>\n<meta property=\"article:published_time\" content=\"2009-11-26T21:59:00+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-1.jpg\" \/>\n<meta name=\"twitter:card\" content=\"summary\" \/>\n<meta name=\"twitter:label1\" content=\"Verfasst von\" \/>\n\t<meta name=\"twitter:data1\" content=\"wp_blogadmin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Gesch\u00e4tzte Lesezeit\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 Minuten\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebSite\",\"@id\":\"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#website\",\"url\":\"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/\",\"name\":\"Merkbar.\",\"description\":\"IT, Elektronik und Mathematik\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"de\"},{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#primaryimage\",\"inLanguage\":\"de\",\"url\":\"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-1.jpg\",\"contentUrl\":\"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-1.jpg\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#webpage\",\"url\":\"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/\",\"name\":\"L\u00f6sung Quiz QUA 2007\/4 - Merkbar.\",\"isPartOf\":{\"@id\":\"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#primaryimage\"},\"datePublished\":\"2009-11-26T21:59:00+00:00\",\"dateModified\":\"2009-11-26T21:59:00+00:00\",\"author\":{\"@id\":\"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#\/schema\/person\/47691942dec3f2eb9d34bb8b5507870d\"},\"breadcrumb\":{\"@id\":\"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#breadcrumb\"},\"inLanguage\":\"de\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Startseite\",\"item\":\"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"L\u00f6sung Quiz QUA 2007\/4\"}]},{\"@type\":\"Person\",\"@id\":\"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#\/schema\/person\/47691942dec3f2eb9d34bb8b5507870d\",\"name\":\"wp_blogadmin\",\"image\":{\"@type\":\"ImageObject\",\"@id\":\"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#personlogo\",\"inLanguage\":\"de\",\"url\":\"http:\/\/1.gravatar.com\/avatar\/d0dc804558b03f640b22e497ec010c9a?s=96&d=mm&r=g\",\"contentUrl\":\"http:\/\/1.gravatar.com\/avatar\/d0dc804558b03f640b22e497ec010c9a?s=96&d=mm&r=g\",\"caption\":\"wp_blogadmin\"},\"url\":\"http:\/\/52.29.166.97\/myblog\/author\/wp_blogadmin\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"L\u00f6sung Quiz QUA 2007\/4 - Merkbar.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/","og_locale":"de_DE","og_type":"article","og_title":"L\u00f6sung Quiz QUA 2007\/4 - Merkbar.","og_description":"Wir nennen das Gef\u00e4ss mit 8 Liter Inhalt A, das mit 5 heisse B und das dritte mit 3 Liter sei C.\u00a0Den F\u00fcllstand k\u00f6nnen mir vereinfacht mit einer 3-stelligen Zahl kodieren, z.B. bedeute\u00a0800 = 8 Liter im Gef\u00e4ss A, 0 Liter in B und 0 in C. Die L\u00f6sung lautet in dieser Notation: 800 \u2192 [&hellip;]","og_url":"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/","og_site_name":"Merkbar.","article_published_time":"2009-11-26T21:59:00+00:00","og_image":[{"url":"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-1.jpg"}],"twitter_card":"summary","twitter_misc":{"Verfasst von":"wp_blogadmin","Gesch\u00e4tzte Lesezeit":"2 Minuten"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebSite","@id":"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#website","url":"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/","name":"Merkbar.","description":"IT, Elektronik und Mathematik","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"de"},{"@type":"ImageObject","@id":"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#primaryimage","inLanguage":"de","url":"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-1.jpg","contentUrl":"http:\/\/wp.andreas.bieri.name\/wp-content\/uploads\/2009\/11\/q07-1.jpg"},{"@type":"WebPage","@id":"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#webpage","url":"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/","name":"L\u00f6sung Quiz QUA 2007\/4 - Merkbar.","isPartOf":{"@id":"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#website"},"primaryImageOfPage":{"@id":"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#primaryimage"},"datePublished":"2009-11-26T21:59:00+00:00","dateModified":"2009-11-26T21:59:00+00:00","author":{"@id":"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#\/schema\/person\/47691942dec3f2eb9d34bb8b5507870d"},"breadcrumb":{"@id":"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#breadcrumb"},"inLanguage":"de","potentialAction":[{"@type":"ReadAction","target":["https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/2009\/11\/26\/lsung-quiz-qua-20074\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Startseite","item":"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/"},{"@type":"ListItem","position":2,"name":"L\u00f6sung Quiz QUA 2007\/4"}]},{"@type":"Person","@id":"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#\/schema\/person\/47691942dec3f2eb9d34bb8b5507870d","name":"wp_blogadmin","image":{"@type":"ImageObject","@id":"http:\/\/ec2-52-29-166-97.eu-central-1.compute.amazonaws.com\/myblog\/#personlogo","inLanguage":"de","url":"http:\/\/1.gravatar.com\/avatar\/d0dc804558b03f640b22e497ec010c9a?s=96&d=mm&r=g","contentUrl":"http:\/\/1.gravatar.com\/avatar\/d0dc804558b03f640b22e497ec010c9a?s=96&d=mm&r=g","caption":"wp_blogadmin"},"url":"http:\/\/52.29.166.97\/myblog\/author\/wp_blogadmin\/"}]}},"_links":{"self":[{"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/posts\/744"}],"collection":[{"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/comments?post=744"}],"version-history":[{"count":0,"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/posts\/744\/revisions"}],"wp:attachment":[{"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/media?parent=744"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/categories?post=744"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/52.29.166.97\/myblog\/wp-json\/wp\/v2\/tags?post=744"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}