{"id":11142,"date":"2023-12-13T10:42:12","date_gmt":"2023-12-13T13:42:12","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11142"},"modified":"2023-12-25T03:56:36","modified_gmt":"2023-12-25T06:56:36","slug":"block-wise-construction-of-bimagic-squares-of-orders-121-and-1331","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=11142","title":{"rendered":"Bimagic Squares of Orders 121 and 1331: Blocks of Order 11"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><strong><em>Whole the work is done manually on excel sheets<\/em><\/strong>.<\/p>\n\n\n\n<p class=\"has-text-align-justify has-electric-grass-gradient-background has-background\">Below are <strong>bimagic squares<\/strong> written in blocks multiples of orders 11. These are of orders 121 and 1331. The work is done manually by author in 2011.  It is summarized in the following link. <br><br><strong>Inder J. Taneja<\/strong>, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011,  pp. 1-14, (22.02.2011), h<a href=\"https:\/\/doi.org\/10.48550\/arXiv.1102.3052\">ttps:\/\/doi.org\/10.48550\/arXiv.1102.3052<\/a>.<\/p>\n\n\n\n<p>Before proceeding further below are the basic formulas to check the sums of <strong>magic<\/strong> and <strong>bimagic<\/strong> squares<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Magic Sum<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"176\" height=\"88\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/S-2.png\" alt=\"\" class=\"wp-image-11140\"\/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Bimagic Sum<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"304\" height=\"93\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2.png\" alt=\"\" class=\"wp-image-11141\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2.png 304w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2-300x92.png 300w\" sizes=\"(max-width: 304px) 100vw, 304px\" \/><\/figure>\n<\/div>\n\n\n<p>where <strong><em>n<\/em><\/strong> is the order the magic square.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 121<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>121&#215;121<\/sub>:=885841; Sb<sub>121&#215;121<\/sub>:= 8646694001<\/strong>, <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Blocks of order 11 are pandiagonal and of equal sums, <strong>S<sub>11&#215;11<\/sub> :=80531<\/strong>. The magic squares of order 121 are also <strong>pandiagonal<\/strong>.  See below the excel sheet for the order 121.<\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 54%,rgb(255,105,0) 100%)\">Excel sheet of Bimagic Square of Order 121<\/h4>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-66201883-062e-49fd-a7ae-37e8265eab79\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/121x121-bi.xlsx\">121&#215;121-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/121x121-bi.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-66201883-062e-49fd-a7ae-37e8265eab79\">Download<\/a><\/div>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 1331: Blocks of Orders 11 and 121<\/h3>\n\n\n\n<p class=\"has-text-align-justify\"> <strong>Bimagic square<\/strong> of order 1331 with magic and bimagic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>1331&#215;1331<\/sub>:=1178974511 and Sb<strong><sub>1331&#215;1331<\/sub><\/strong>:= 1392417235445951<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify\">Block of order 121 are also <strong>pandiagonal bimagic<\/strong> square with different bimagic, but equal magic sums, <strong>S<sub>121&#215;121<\/sub>:=107179501.<\/strong>  Block of order 11 are also pandiagonal with equal sums <strong>S<sub>11&#215;11<\/sub>:=9743591.<\/strong> <\/p>\n\n\n\n<p class=\"has-text-align-justify\">The magic square of order 1331 is also <strong>pandiagonal<\/strong>. It is checked with software by <a href=\"https:\/\/budshaw.ca\/Download.html\">H. White<\/a>.  See below excel sheet of the bimagic square of order 1331.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 54%,rgb(255,105,0) 100%)\">Excel sheet of Bimagic Square of Order 1331<\/h4>\n\n\n\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-5131e63a-cd24-43f1-95d0-45b02c4a5667\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/1331x1331-bi.xlsx\">1331&#215;1331-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/1331x1331-bi.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-5131e63a-cd24-43f1-95d0-45b02c4a5667\">Download<\/a><\/div>\n\n\n\n<p class=\"has-text-align-justify\">Due to higher values in the excel the the bimagic sum is appearing as <strong>Sb<strong><sub>1331&#215;1331<\/sub><\/strong>:= 1392417235445950. <\/strong>Actually it is <strong><strong>Sb<strong><sub>1331&#215;1331<\/sub><\/strong>:= 1392417235445951<\/strong><\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Whole the work is done manually on excel sheets. Below are bimagic squares written in blocks multiples of orders 11. These are of orders 121 and 1331. The work is done manually by author in 2011. It is summarized in the following link. Inder J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3],"tags":[],"class_list":["post-11142","post","type-post","status-publish","format-standard","hentry","category-magic-squares"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11142","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11142"}],"version-history":[{"count":2,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11142\/revisions"}],"predecessor-version":[{"id":11225,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11142\/revisions\/11225"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11142"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11142"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11142"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}