Below are bimagic squares<\/strong> written in blocks multiples of orders 16. This work brings magic squares multiples of 16 and 64, i.e., magic squares of orders 256, 512 and 1024. Most of the work is done manually by author in 2011. See the reference below: Before proceeding further below are the basic formulas to check the sums of magic<\/strong> and bimagic<\/strong> squares<\/p>\n\n\n\n where n<\/em><\/strong> is the order the magic square.<\/p>\n\n\n\n Before giving the magic squares of orders 256, 512 and 1025, below are examples of bimagic squares of orders 16 and 64.<\/p>\n\n\n\n In this case, we have<\/p>\n\n\n\n S16×16<\/sub> := 2056 and Sb16×16<\/sub> := 351576, <\/strong><\/p>\n\n\n\n See below the bimagic square<\/strong> of order 16.<\/p>\n\n\n Magic squares of order 4 are of equal sums, S4×4<\/sub> :=(1\/4) S16×16<\/sub>=514.<\/strong><\/p>\n\n\n\n <\/p>\n\n\n\n In this case, we have<\/p>\n\n\n\n S64×64<\/sub> :=131104 and Sb64×64<\/sub> := 358045024.<\/strong> <\/p>\n\n\n\n See below bimagic square<\/strong> of order 64 with blocks of order 16. These blocks of order 16 are bimagic squares with different bimagic sums, but the magic sums of all order 16 magic squares are same, i.e., S16×16<\/sub> := 32776. <\/strong>Moreover, magic sums of magic squares of order 4 are also same, i.e., S4×4<\/sub> := 8194<\/strong><\/strong><\/strong><\/p>\n\n\n <\/p>\n\n\n\n Bimagic square of order 256×256<\/strong> has the the following properties:<\/p>\n\n\n\n <\/p>\n\n\n\n Summarizing, we have a bimagic square of order 256×256<\/strong>, where blocks of orders 4×4, 16×16 and 64×64 are of equal magic sums. The magic squares of orders 16×16 bimagic with different bimagic sums. The magic squares of order 64×64 are of equal bimagic sums<\/strong>. These values are given in tables in an excel sheet attached with the work.<\/p>\n\n\n\n <\/p>\n\n\n\n Bimagic square of order 512×512<\/strong> has the the following properties:<\/p>\n\n\n\n <\/p>\n\n\n\n Summarizing, we have a bimagic square of order 512×512<\/strong>, where blocks of orders 4×4, 16×16, 128×128 are of equal magic sums. The magic squares of orders 16×16 and 128×128 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 <\/p>\n\n\n\n <\/p>\n\n\n\n It is bimagic square of order 1024×1024 with the following properties:<\/p>\n\n\n\n <\/p>\n\n\n\n Summarizing, we have a bimagic square of order 1024×1024<\/strong>, where blocks of orders 4×4, 16×16, 64×64 are and 128×128 are of equal magic sums. The magic squares of orders 16×16, 64×64 and 256×256 are bimagic with different bimagic sums<\/strong>. Each block of order 256×256, the bimagic sums of order 64×64 are the same. These values are given in tables in an excel sheet attached with the work.<\/p>\n\n\n\n <\/p>\n\n\n\n Whole the work is done manually on excel sheets. Below are bimagic squares written in blocks multiples of orders 16. This work brings magic squares multiples of 16 and 64, i.e., magic squares of orders 256, 512 and 1024. Most of the work is done manually by author in 2011. See the reference below: Inder […]<\/p>\n","protected":false},"author":1,"featured_media":11088,"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-11168","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-1.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11168","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=11168"}],"version-history":[{"count":7,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11168\/revisions"}],"predecessor-version":[{"id":11195,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11168\/revisions\/11195"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/11088"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11168"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11168"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Inder J. Taneja<\/strong>, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011, pp. 1-14, (22.02.2011), https:\/\/doi.org\/10.48550\/arXiv.1102.3052<\/a>.<\/p>\n\n\n\n\n
![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/S-5.png\")
<\/figure>\n<\/div>\n\n\n\n
![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-5.png\")
<\/figure>\n<\/div>\n\n\nBimagic Square of Order 16<\/h3>\n\n\n\n
![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2.png\")
<\/figure>\n<\/div>\n\n\nBimagic Square of Order 64: Blocks of Order 16<\/h3>\n\n\n\n
![\"\"](\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1.png\")
<\/figure>\n<\/div>\n\n\nBimagic Square of Order 256: Blocks of Order 16<\/h3>\n\n\n\n
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Bimagic Square of Order 512: Blocks of Order 16<\/h3>\n\n\n\n
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Bimagic Square of Order 1024: Blocks of Order 1024<\/h3>\n\n\n\n
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References:<\/h3>\n\n\n\n
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