{"id":11142,"date":"2023-12-13T10:42:12","date_gmt":"2023-12-13T13:42:12","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11142"},"modified":"2026-05-19T18:09:31","modified_gmt":"2026-05-19T21:09:31","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 wp-block-paragraph\"><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 wp-block-paragraph\">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 class=\"wp-block-paragraph\">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 class=\"wp-block-paragraph\">where <strong><em>n<\/em><\/strong> is the order the magic square.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/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 class=\"wp-block-paragraph\">In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><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 wp-block-paragraph\">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 wp-block-paragraph\"> <strong>Bimagic square<\/strong> of order 1331 with magic and bimagic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><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 wp-block-paragraph\">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 wp-block-paragraph\">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 class=\"wp-block-paragraph\"><\/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 wp-block-paragraph\">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\n\n\n<p class=\"wp-block-paragraph\"><\/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%)\">References<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><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>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11031\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares: Multiples of Orders 8 and 16<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11090\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares Multiples of 25: Orders 25, 125 and 625<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11120\">Block-Wise Construction of Bimagic Squares of Orders 49 and 343.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11129\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares Multiples of 9: Orders 9, 81 and 729<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11142\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares of Orders 121 and 1331<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11168\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 256, 512 and 1024: Blocks of Order 16.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11186\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 200 and 1000: Blocks of Order 8.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11198\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 400, 800, 1600 and 2000: Blocks of Order 16.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11224\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 100, 110 and 121: Blocks of Orders 10 and 11.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=13138\">Universal and Upside-Down Magic and Bimagic Squares of Order 16<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=13336\">Universal and Upside-Down Magic and Bimagic Squares of Order 25<\/a><\/li>\n<\/ol>\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_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"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":3,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11142\/revisions"}],"predecessor-version":[{"id":18909,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11142\/revisions\/18909"}],"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}]}}