{"id":11198,"date":"2023-12-23T23:27:24","date_gmt":"2023-12-24T02:27:24","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11198"},"modified":"2026-05-19T18:13:45","modified_gmt":"2026-05-19T21:13:45","slug":"bimagic-squares-of-orders-400-800-1600-and-2000-blocks-of-order-16","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=11198","title":{"rendered":"Bimagic Squares of Orders 400, 800, 1600 and 2000: Blocks of Order 16"},"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 16. This work brings bimagic squares multiples of 16, such as, of orders 400, 800, 1600 and 2000. Most of 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-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>where <strong><em>n<\/em><\/strong> is the order the magic square.<\/p>\n\n\n\n<p class=\"has-text-align-justify\">Before presenting <strong>bimagic squares<\/strong> of orders 400, 800, 1600 and 2000, below is an example of bimagic square of order 16.<\/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 16<\/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>16&#215;16<\/sub> := 2056 and Sb<sub>16&#215;16<\/sub> := 351576, <\/strong><\/p>\n\n\n\n<p>See below the <strong>bimagic square<\/strong> of order 16.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"1700\" height=\"758\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2.png\" alt=\"\" class=\"wp-image-11106\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2.png 1700w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-300x134.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-1024x457.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-768x342.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-535x239.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-1536x685.png 1536w\" sizes=\"(max-width: 1700px) 100vw, 1700px\" \/><\/figure>\n<\/div>\n\n\n<p>Magic squares of order 4 are of equal sums, <strong>S<sub>4&#215;4<\/sub> :=(1\/4) S<sub>16&#215;16<\/sub>=514.<\/strong><\/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 400: Blocks of Order 16<\/h3>\n\n\n\n<p><strong>Bimagic square of order 400&#215;400<\/strong> has the the following properties:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>4&#215;4 are <strong>magic squares<\/strong> with <strong>equal magic sums<\/strong>, i.e., <strong>S<sub>4&#215;4<\/sub>:=320002<\/strong><\/li>\n\n\n\n<li>16&#215;16 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>16&#215;16<\/sub>:=1280008<\/strong>. <\/li>\n\n\n\n<li>80&#215;80 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>80&#215;80<\/sub>:=6400040<\/strong>.<\/li>\n\n\n\n<li>400&#215;400 is a <strong>bimagic square<\/strong> with <strong>magic and bimagic sums<\/strong>, i.e., <strong>S<sub>256&#215;256<\/sub>:=32000200 <\/strong>and<strong> <strong>Sb<sub>256&#215;256<\/sub>:=3413365333400<\/strong>.<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"has-text-align-justify has-pale-ocean-gradient-background has-background\">Summarizing, we have a <strong>bimagic square of order 400&#215;400<\/strong>, where blocks of orders 4&#215;4, 16&#215;16 and 80&#215;80 are of equal magic sums. The magic squares of orders 16&#215;16 and 80&#215;80 are bimagic with different bimagic sums. These values are given in tables in an excel sheet attached with the work.<\/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-fb9fb050-9394-4965-a3aa-d24dbb032fd7\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/400x400-16x16-4x4-Final.xlsx\">400&#215;400-16&#215;16-4&#215;4-Final<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/400x400-16x16-4x4-Final.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-fb9fb050-9394-4965-a3aa-d24dbb032fd7\">Download<\/a><\/div>\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 800: Blocks of Order 16<\/h3>\n\n\n\n<p><strong>Bimagic square of order 800&#215;800<\/strong> has the the following properties:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>4&#215;4 are <strong>magic squares<\/strong> with <strong>equal magic sums<\/strong>, i.e., <strong>S<sub>4&#215;4<\/sub>:=1280002<\/strong><\/li>\n\n\n\n<li>16&#215;16 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>16&#215;16<\/sub>:=5120008<\/strong>. <\/li>\n\n\n\n<li>160&#215;160 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>160&#215;160<\/sub>:=51200080<\/strong>.<\/li>\n\n\n\n<li>800&#215;800 is a <strong>bimagic square<\/strong> with <strong>magic and bimagic sums<\/strong>, i.e., <strong>S<sub>800&#215;800<\/sub>:=256000400 <\/strong>and<strong> <strong>Sb<sub>800&#215;800<\/sub>:=109226922666800<\/strong>.<\/strong><\/li>\n<\/ol>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-pale-ocean-gradient-background has-background\">Summarizing, we have a <strong>bimagic square of order 800&#215;800<\/strong>, where blocks of orders 4&#215;4, 16&#215;16 and 160&#215;160 are of equal magic sums. The magic squares of orders 16&#215;16 and 160&#215;160 are bimagic with different bimagic sums. These values are given in tables in an excel sheet attached with the work.<\/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-be28b79a-19d1-41a6-8de7-c551fb589de8\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/800x800-16x16-4x4-Final.xlsx\">800&#215;800-16&#215;16-4&#215;4-Final<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/800x800-16x16-4x4-Final.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-be28b79a-19d1-41a6-8de7-c551fb589de8\">Download<\/a><\/div>\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 1600: Blocks of Order 16<\/h3>\n\n\n\n<p><strong>Bimagic square of order 1600&#215;1600<\/strong> has the the following properties:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>4&#215;4 are <strong>magic squares<\/strong> with <strong>equal magic sums<\/strong>, i.e., <strong>S<sub>4&#215;4<\/sub>:=5120002<\/strong><\/li>\n\n\n\n<li>16&#215;16 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>16&#215;16<\/sub>:=20480008<\/strong>. <\/li>\n\n\n\n<li>80&#215;80 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>80&#215;80<\/sub>:=102400040<\/strong>.<\/li>\n\n\n\n<li>400&#215;400 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>400&#215;400<\/sub>:=<\/strong>512000200.<\/li>\n\n\n\n<li>1600&#215;1600 is a <strong>bimagic square<\/strong> with <strong>magic and bimagic sums<\/strong>, i.e., <strong>S<sub>1600&#215;1600<\/sub>:=2048000800 <\/strong>and<strong> <strong>Sb<sub>1600&#215;1600<\/sub>:=3495255381333600<\/strong>.<\/strong><\/li>\n<\/ol>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-pale-ocean-gradient-background has-background\">Summarizing, we have a <strong>bimagic square of order 1600&#215;1600<\/strong>, where blocks of orders 4&#215;4, 16&#215;16, 80&#215;80 and 400&#215;400 are of equal magic sums. The magic squares of orders 16&#215;16, 80&#215;80 and 400&#215;400 are bimagic with different bimagic sums. These values are given in tables in an excel sheet attached with the work.<\/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-4863576e-5f99-41d2-95b2-92b57707c5d6\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/1600x1600-16x16-4x4-Final.xlsx\">1600&#215;1600-16&#215;16-4&#215;4-Final<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/1600x1600-16x16-4x4-Final.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-4863576e-5f99-41d2-95b2-92b57707c5d6\">Download<\/a><\/div>\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 2000: Blocks of Order 16<\/h3>\n\n\n\n<p><strong>Bimagic square of order 2000&#215;2000<\/strong> has the the following properties:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>4&#215;4 are <strong>magic squares<\/strong> with <strong>equal magic sums<\/strong>, i.e., <strong>S<sub>4&#215;4<\/sub>:=8000002<\/strong><\/li>\n\n\n\n<li>16&#215;16 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>16&#215;16<\/sub>:=32000008<\/strong>. <\/li>\n\n\n\n<li>80&#215;80 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>80&#215;80<\/sub>:=160000040<\/strong>.<\/li>\n\n\n\n<li>400&#215;400 are <strong>bimagic squares<\/strong> with <strong>equal magic <\/strong>and <strong>different bimagic sums<\/strong>, i.e., <strong>S<sub>400&#215;400<\/sub>:=800000200<\/strong>.<\/li>\n\n\n\n<li>1600&#215;1600 is a <strong>bimagic square<\/strong> with <strong>magic and bimagic sums<\/strong>, i.e., <strong>S<sub>2000&#215;2000<\/sub>:=4000001000 <\/strong>and<strong> <strong>Sb<sub>2000&#215;2000<\/sub>:=<\/strong><\/strong>10666670666666800<strong>.<\/strong><\/li>\n<\/ol>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-pale-ocean-gradient-background has-background\">Summarizing, we have a <strong>bimagic square of order 2000&#215;2000<\/strong>, where blocks of orders 4&#215;4, 16&#215;16, 80&#215;80 and 400&#215;400 are of equal magic sums. The magic squares of orders 16&#215;16, 80&#215;80 and 400&#215;400 are bimagic with different bimagic sums. These values are given in tables in an excel sheet attached with the work.<\/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-862a86bd-da8c-4d4f-840e-b099f4a92814\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/2000x2000-16x16-4x4-Final.xlsx\">2000&#215;2000-16&#215;16-4&#215;4-Final<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/2000x2000-16x16-4x4-Final.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-862a86bd-da8c-4d4f-840e-b099f4a92814\">Download<\/a><\/div>\n\n\n\n<p><\/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%)\">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><\/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 16. This work brings bimagic squares multiples of 16, such as, of orders 400, 800, 1600 and 2000. Most of the work is done manually by author in 2011. See the reference below: Inder J. Taneja, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":11106,"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-11198","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\/16x16-bi-2.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11198","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=11198"}],"version-history":[{"count":5,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11198\/revisions"}],"predecessor-version":[{"id":18912,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11198\/revisions\/18912"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/11106"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11198"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11198"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11198"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}