{"id":11090,"date":"2023-12-12T19:25:51","date_gmt":"2023-12-12T22:25:51","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11090"},"modified":"2023-12-13T10:52:50","modified_gmt":"2023-12-13T13:52:50","slug":"block-wise-construction-of-bimagic-squares-multiples-of-25-orders-25-125-and-625","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=11090","title":{"rendered":"Block-Wise Construction of Bimagic Squares Multiples of 25: Orders 25, 125 and 625"},"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 order 25. These are orders 25, 125 and 625 The work is done manually by author in 2011. See the reference 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>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-3.png\" alt=\"\" class=\"wp-image-11151\"\/><\/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-3.png\" alt=\"\" class=\"wp-image-11153\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-3.png 304w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-3-300x92.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 25<\/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>25&#215;25<\/sub>:=7825; Sb<sub>25&#215;25<\/sub>:= 3263025<\/strong>, <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Blocks of order 5 are pandiagonal and of equal sums, <strong>S<sub>5&#215;5<\/sub> := 1565<\/strong>. The magic squares of order 25 is also pandiagonal. See below the <strong>bimagic square<\/strong> of order 25.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"3061\" height=\"1086\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/25x25-bi.png\" alt=\"\" class=\"wp-image-11111\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/25x25-bi.png 3061w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/25x25-bi-300x106.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/25x25-bi-1024x363.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/25x25-bi-768x272.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/25x25-bi-535x190.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/25x25-bi-1536x545.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/25x25-bi-2048x727.png 2048w\" sizes=\"(max-width: 3061px) 100vw, 3061px\" \/><\/figure>\n<\/div>\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 125: Blocks of Order 25<\/h3>\n\n\n\n<p class=\"has-text-align-justify\">Bimagic Square of order 125&#215;125 with magic and bimagic sums<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>125&#215;125<\/sub>:=976625 and Sb<sub>125&#215;125<\/sub>:= 1073502625<\/strong><\/p>\n\n\n\n<p>Each block of order 25 is a <strong>pandiagonal bimagic<\/strong> square with different bimagic sums, <strong>S<sub>25&#215;25<\/sub>:=195325.<\/strong><\/p>\n\n\n\n<p>Using <a href=\"https:\/\/budshaw.ca\/Download.html\">H. White&#8217;s software<\/a> we can easily check that the bimagic square of order 125 is <strong>pandiagonal<\/strong>.<\/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 125<\/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\/125x125-5x5-1.xlsx\">125&#215;125-5&#215;5-1<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/125x125-5x5-1.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 625: Blocks of Order 25 and 125<\/h3>\n\n\n\n<p class=\"has-text-align-justify\">Bimagic Square of order 625&#215;625 with magic and bimagic sums<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>625&#215;625<\/sub>:=122070625 and Sb<sub>625&#215;625<\/sub>:= 31789265950625<\/strong><\/p>\n\n\n\n<p>Each block of order 125 is a pandiagonal bimagic square with equal magic and bimagic sums<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>125&#215;125<\/sub>:=24414125 and Sb<sub>125&#215;125<\/sub>:= 6357853190125<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify\">Each block of order 25&#215;25 is bimagic square&nbsp;of same magic sums, <strong>S<sub>25&#215;25<\/sub>:=4882825<\/strong>, but different bimagic sums given in a table in excel sheet attached with this work.<\/p>\n\n\n\n<p>Blocks of order 5 are also pandiagonal magic squares with equal magic sums <strong>S<sub>5&#215;5<\/sub>:=(1\/5)S<sub>25&#215;25<\/sub>:=976565<\/strong>.<\/p>\n\n\n\n<p>Using <a href=\"https:\/\/budshaw.ca\/Download.html\">H. White&#8217;s software<\/a>, we can easily check that the bimagic square of order 625 is <strong>pandiagonal<\/strong>.<\/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 625<\/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\/625x625-bi.xlsx\">625&#215;625-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/625x625-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","protected":false},"excerpt":{"rendered":"<p>Whole the work is done manually on excel sheets. Below are bimagic squares written in blocks multiples of order 25. These are orders 25, 125 and 625 The work is done manually by author in 2011. See the reference below: Inder J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":11112,"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-11090","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\/25x25-bi-1.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11090","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=11090"}],"version-history":[{"count":7,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11090\/revisions"}],"predecessor-version":[{"id":11154,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11090\/revisions\/11154"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/11112"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11090"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11090"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11090"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}