{"id":133,"date":"2018-02-20T09:57:16","date_gmt":"2018-02-20T15:57:16","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?page_id=133"},"modified":"2018-03-26T14:11:15","modified_gmt":"2018-03-26T19:11:15","slug":"problem-of-the-week-problem-4-solution-spring-2018","status":"publish","type":"page","link":"https:\/\/blogs.acu.edu\/mu_sigma\/problem-of-the-week-problem-4-solution-spring-2018\/","title":{"rendered":"Problem of the Week &#8211; Problem 4 Solution &#8211; February 22, 2018"},"content":{"rendered":"<p><span style=\"font-size: large\">Solution to <b>An Unmagic Square?<\/b><\/span><\/p>\n<p>Correct Solutions: \u00a0<span style=\"text-decoration: underline\">No correct solutions were submitted<\/span>.<\/p>\n<p>A 3&#215;3 matrix with all its entries from the set {-1, 0, 1} cannot have the sum of all its rows and columns to be distinct.<\/p>\n<p>For suppose such a matrix could be constructed. The possible sums of the rows and columns are -3, -2, -1, 0, 1, 2, and 3. So six of these seven numbers would have to be used. Here are some facts:<\/p>\n<ul>\n<li style=\"list-style-type: none\">\n<ul>\n<li>The numbers 3 and -3 cannot both be used. If they could, two of the rows (or columns) would have to be [1 1 1] and [-1 -1 -1]. Then, in order to make the sum of the three columns distinct, the third row would have to be some permutation of [-1 0 1]. But then the sum of this row and the sum of the column containing the 0 are both zero. This is a contradiction to all of the sums&#8217; being distinct.<\/li>\n<li>Without loss of generality, assume -3 is the only missing sum. The numbers 3 = 1 + 1 + 1 , 2 = 1 + 1 + 0, and -2 = (-1)+ (-1) + 0 have unique (except for order of addition) representations as sums of the numbers -1, 0, and 1. The numbers -1, 0, and 1 can be expressed in more than one way by summing the numbers -1, 0, and 1.<\/li>\n<li>Since the numbers 3 and -2 must be sums of a row or a column, we assume row 1 of the matrix is [1 1 1] and row 2 is [-1 -1 0]. (It is clear that we cannot have one of these as a row and the other as a column.) Since we must have some row or column with a sum of 2, it follows that the last row of the matrix must be of the form [* * 1], where the two starred positions are yet to be determined.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>To finish the proof, try each of three cases in turn: First try -1 in the southwest corner of the martix; then try 0 in the southwest corner; and finally try 1 in the southwest corner. None of these cases work.<\/p>\n<p>Therefore the proposed matrix cannot be constructed.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution to An Unmagic Square? Correct Solutions: \u00a0No correct solutions were submitted. A 3&#215;3 matrix with all its entries from the set {-1, 0, 1} cannot have the sum of all its rows and columns to be distinct. For suppose such a matrix could be constructed. The possible sums of the rows and columns are [&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-133","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/133","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=133"}],"version-history":[{"count":3,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/133\/revisions"}],"predecessor-version":[{"id":144,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/pages\/133\/revisions\/144"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}