{"id":103,"date":"2017-10-30T14:42:48","date_gmt":"2017-10-30T19:42:48","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?page_id=103"},"modified":"2017-10-30T14:44:10","modified_gmt":"2017-10-30T19:44:10","slug":"problem-of-the-week-problem-1-solution-october-16-2017","status":"publish","type":"page","link":"https:\/\/blogs.acu.edu\/mu_sigma\/problem-of-the-week-problem-1-solution-october-16-2017\/","title":{"rendered":"Problem of the Week &#8211; Problem 1 Solution &#8211; October 16, 2017"},"content":{"rendered":"<p><span style=\"font-size: xx-large\"><b>How Many Rectangles?<\/b><\/span><\/p>\n<p><i>This problem was provided by Michael Khoury, US Math Olympiad Team Member.<\/i><\/p>\n<p>How many rectangles (of any size) are present on a standard 8&#215;8 chessboard?<\/p>\n<p><span style=\"font-size: small\">Correct solutions were submitted by: \u00a0Dakotah Martinez, Bethany Witemeyer<\/span><\/p>\n<p>Choosing a rectangle is equivalent to choosing the four lines that bound it. There are nine lines running in each direction so there are <sub>9<\/sub>C<sub>2<\/sub> = 36 ways to choose the horizontal borders. (<sub>n<\/sub>C<sub>r<\/sub> is the binomial coefficient, out of <i>n<\/i> choose <i>r<\/i> where order of selection does not matter.) Similarly there are 36 ways to choose the vertical borders. The total number of rectangles is therefore 36<sup>2<\/sup> = 1296.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>How Many Rectangles? This problem was provided by Michael Khoury, US Math Olympiad Team Member. How many rectangles (of any size) are present on a standard 8&#215;8 chessboard? Correct solutions were submitted by: \u00a0Dakotah Martinez, Bethany Witemeyer Choosing a rectangle is equivalent to choosing the four lines that bound it. There are nine lines running [&hellip;]<\/p>\n","protected":false},"author":130,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"class_list":["post-103","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/103","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=103"}],"version-history":[{"count":1,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/103\/revisions"}],"predecessor-version":[{"id":104,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/103\/revisions\/104"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=103"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}