{"id":11186,"date":"2023-12-23T17:33:18","date_gmt":"2023-12-23T20:33:18","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11186"},"modified":"2026-05-19T18:12:27","modified_gmt":"2026-05-19T21:12:27","slug":"bimagic-squares-of-orders-200-and-1000-multiples-of-order-8","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=11186","title":{"rendered":"Bimagic Squares of Orders 200 and 1000: Blocks of Order 8"},"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 8. These are bimagic squares of order 200 and 1000. Most of the work is done manually by author in 2011. <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-5.png\" alt=\"\" class=\"wp-image-11160\"\/><\/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-5.png\" alt=\"\" class=\"wp-image-11161\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-5.png 304w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-5-300x92.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 8<\/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>8&#215;8<\/sub>:=260; Sb<sub>8&#215;8<\/sub>:= 11180<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">2&#215;4 blocks are of equal sums as of magic square, i.e., 260. See below the <strong>bimagic square<\/strong> of order 8.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"780\" height=\"433\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1.png\" alt=\"\" class=\"wp-image-11079\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1.png 780w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1-300x167.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1-768x426.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1-535x297.png 535w\" sizes=\"(max-width: 780px) 100vw, 780px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><em>The construction of <\/em>bimagic<em> square of order 8 is well known in the history, and is done by G. Pfeffermann in 1891. In this case, we have a <strong>pandiagonal bimagic<\/strong> square of order 8, where the blocks of order 2&#215;4 are of same sum as of magic square of order 8.<\/em><\/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 40: Blocks of Order 8<\/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>40&#215;40<\/sub>:=32020; Sb<sub>40&#215;40<\/sub>:= 34165340, S<sub>8&#215;8<\/sub>:=6404.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">Magic squares of order 8 are of equal sums. These are either bimagic or semi-bimagic. See <strong>bimagic square<\/strong> of order 40. It is also pandiagonal.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2701\" height=\"1329\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi.png\" alt=\"\" class=\"wp-image-11064\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi.png 2701w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-300x148.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-1024x504.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-768x378.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-535x263.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-1536x756.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-2048x1008.png 2048w\" sizes=\"(max-width: 2701px) 100vw, 2701px\" \/><\/figure>\n<\/div>\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 200: Blocks of Orders 8 and 40<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Bimagic square of order 200\u00d7200<\/strong>&nbsp;has the the following properties:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>8\u00d78 are&nbsp;<strong>pandiagonal magic squares<\/strong>&nbsp;with&nbsp;<strong>equal magic sums<\/strong>, i.e.,&nbsp;<strong>S<sub>8&#215;8<\/sub>:=160004<\/strong>. These are with different bimagic or semi-bimagic sums.<\/li>\n\n\n\n<li>40\u00d740 are&nbsp;<strong>pandiagonal equal sums magic squares<\/strong> with different <strong>bimagic sums, <\/strong>i.e., <strong> S<sub>40\u00d740<\/sub>:=800020<\/strong>. These are with different bimagic or semi-bimagic sums.<\/li>\n\n\n\n<li>200&#215;200 is a <strong>pandiagonal bimagic squares<\/strong>&nbsp;with&nbsp;<strong>magic and bimagic sums S<sub>200\u00d7200<\/sub>:=4000100 <\/strong>and  <strong>Sb<sub>200&#215;200<\/sub>:=106670666700<\/strong>.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-blush-light-purple-gradient-background has-background wp-block-paragraph\">Summarizing, we have a&nbsp;<strong>pandiagonal bimagic square of order 200\u00d7200<\/strong>, where blocks of orders 8\u00d78 and 40&#215;40 are also pandiagonal with equal magic sums, but different bimagic or semi-bimagic sums. These values are given in tables in an excel sheet attached with the work.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Excel file for download<\/strong><\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-01c04a03-ff28-4e9b-a333-6c79fa0af96c\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/200x200-8x8-Final.xlsx\">200&#215;200-8&#215;8-Final<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/200x200-8x8-Final.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-01c04a03-ff28-4e9b-a333-6c79fa0af96c\">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%)\">Bimagic Square of Order 1000: Blocks of Orders 8, 40 and 200<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Bimagic square of order 1000\u00d71000<\/strong>&nbsp;has the the following properties:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>8\u00d78 are&nbsp;<strong>pandiagonal equal sums magic squares<\/strong>, i.e.,&nbsp;<strong>S<sub>8&#215;8<\/sub>:=4000004<\/strong>. These are with different bimagic or semi-bimagic sums.<\/li>\n\n\n\n<li>40\u00d740 are&nbsp;<strong>pandiagonal equal sums magic squares<\/strong>, i.e., <strong>S<sub>40\u00d740<\/sub>:=20000020<\/strong>. These are with different bimagic or semi-bimagic sums.<\/li>\n\n\n\n<li>200&#215;200 are pandiagonal <strong>equal sum magic squares<\/strong>, i.e, <strong><strong>S<sub>200\u00d7200<\/sub>:=<\/strong>100000100. <\/strong>These are with different bimagic or semi-bimagic sums.<\/li>\n\n\n\n<li>1000&#215;1000 is a <strong>pandiagonal bimagic square<\/strong> with magic and bimagic sums, <strong>S<sub>1000\u00d71000<\/sub>:=500000500<\/strong> and <strong><strong>Sb<sub>1000\u00d71000<\/sub><\/strong>:=333333833333500<\/strong>.<\/li>\n<\/ol>\n\n\n\n<p class=\"has-text-align-justify has-blush-light-purple-gradient-background has-background wp-block-paragraph\">Summarizing, we have a&nbsp;<strong>pandiagonal bimagic square of order 1000\u00d71000<\/strong>, where blocks of orders 8\u00d78, 40&#215;40 and 200&#215;200 are also pandiagonal with equal magic sums, but different bimagic or semi-bimagic sums. These values are given in tables in an excel sheet attached with the work.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Excel file for download<\/strong><\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-150d7f10-4e5b-4d7e-9703-62e4717b8b9a\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/1000x1000-8x8-Final.xlsx\">1000&#215;1000-8&#215;8-Final<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/1000x1000-8x8-Final.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-150d7f10-4e5b-4d7e-9703-62e4717b8b9a\">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","protected":false},"excerpt":{"rendered":"<p>Whole the work is done manually on excel sheets. Below are bimagic squares written in blocks multiples of orders 8. These are bimagic squares of order 200 and 1000. Most of the work is done manually by author in 2011. Inder J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":11190,"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-11186","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\/8X8-bi-2.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11186","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=11186"}],"version-history":[{"count":5,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11186\/revisions"}],"predecessor-version":[{"id":18911,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11186\/revisions\/18911"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/11190"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11186"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11186"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}