{"id":11129,"date":"2023-12-13T05:08:58","date_gmt":"2023-12-13T08:08:58","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11129"},"modified":"2026-05-19T18:05:14","modified_gmt":"2026-05-19T21:05:14","slug":"block-wise-construction-of-bimagic-squares-multiples-of-9-orders-9-81-and-729","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=11129","title":{"rendered":"Block-Wise Construction of Bimagic Squares Multiples of 9: Orders 9, 81 and 729"},"content":{"rendered":"\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 order 9. These are of orders 9, 81 and 729 The work is done manually by author in 2011.  The work is summarized in the link below: <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-1.png\" alt=\"\" class=\"wp-image-11132\"\/><\/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-1.png\" alt=\"\" class=\"wp-image-11133\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-1.png 304w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-1-300x92.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 9<\/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>9&#215;9<\/sub>:=369; Sb<sub>9&#215;9<\/sub>:= 20049<\/strong>, <\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">Sum of entries of blocks of order 3&#215;3 are the same as of magic square. i.e., <strong>S<sub>9&#215;9<\/sub>:=369<\/strong>. Below is an example of bimagic square of order 9. It is constructed by G. Pfeffermann in 1891.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"982\" height=\"547\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/9x9-bi.png\" alt=\"\" class=\"wp-image-11134\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/9x9-bi.png 982w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/9x9-bi-300x167.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/9x9-bi-768x428.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/9x9-bi-535x298.png 535w\" sizes=\"(max-width: 982px) 100vw, 982px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">We shall use the idea of this bimagic square to bring bimagic squares of orders 81 and 729.<\/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 81: Blocks of Order 9<\/h3>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><strong>Bimagic Square<\/strong> of order 81&#215;81 with magic and bimagic sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>81&#215;81<\/sub>:=265761 and Sb<strong><sub>81&#215;81<\/sub><\/strong>:=1162527201 <\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">Each block of order 9 is a <strong>magic<\/strong> square with equal magic sums, <strong>S<sub>9&#215;9<\/sub>:=<\/strong>29529. It also has the same property as of order 9, i.e., sum of entries of blocks of order 3&#215;3 are the same as of magic square of order 9, i.e., <strong>S<sub>9&#215;9<\/sub>:=<\/strong>29529.<\/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 81<\/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\/81x81-bi.xlsx\">81&#215;81-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/81x81-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 729: Blocks of Order 9 and 81<\/h3>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">Bimagic Square of order 729 with magic and bimagic sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>729&#215;729<\/sub> :=193710609 and Sb<strong><sub>729&#215;729<\/sub><\/strong><\/strong> <strong>:= 68630571075249<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of orders 9 and 81 are only magic squares with equal magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>81&#215;81<\/sub> := 21523401 and S<sub>9&#215;9<\/sub> := 2391489<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Similar to magic squares of orders 9 and 81 as given above, in this case also the sum of entries of blocks of order 3&#215;3 are the same as of magic square of orderr 9, i.e., <strong>S<sub>9&#215;9<\/sub> := 2391489<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>Unfortunately, in this case, we don&#8217;t have sub-groups of orders 9 or 81 as bimagic squares. These are only magic squares of equal sums.<\/strong><\/em><\/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 729<\/h4>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-d41e8160-b388-44b5-a1d0-425c8060a4f5\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/729x729-bi.xlsx\">729&#215;729-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/729x729-bi.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-d41e8160-b388-44b5-a1d0-425c8060a4f5\">Download<\/a><\/div>\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\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Below are bimagic squares written in blocks multiples of order 9. These are of orders 9, 81 and 729 The work is done manually by author in 2011. The work is summarized in the link below: Inder J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011, pp. 1-14, (22.02.2011), [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":11134,"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-11129","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-magic-squares"],"jetpack_featured_media_url":"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/9x9-bi.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11129","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=11129"}],"version-history":[{"count":4,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11129\/revisions"}],"predecessor-version":[{"id":18907,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11129\/revisions\/18907"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/11134"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11129"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11129"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}