{"id":654,"date":"2022-02-10T21:49:26","date_gmt":"2022-02-11T00:49:26","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=654"},"modified":"2022-02-17T10:21:39","modified_gmt":"2022-02-17T13:21:39","slug":"area-representations-magic-squares-with-fraction-numbers-entries","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=654","title":{"rendered":"Area Representations Magic Squares With Fraction Numbers Entries"},"content":{"rendered":"\n<p class=\"has-text-align-justify\">It is well known that every magic square can be written as&nbsp;<strong>perfect square sum of entries<\/strong>. It is always possible with odd number entries starting from 1. In case of odd order magic squares we can also write with&nbsp;<strong>consecutive natural number<\/strong>&nbsp;entries. Still, it is unknown whether it is possible to even order magic squares. In case of odd order magic squares, still we can write them with&nbsp;<strong>minimum perfect square sum<\/strong>&nbsp;of entries. Based on this idea of&nbsp;<strong>perfect square sum of entries<\/strong>, we have written a magic square representing areas. It is done for the magic squares of orders 3 to 11. In all the cases, the area representations are more that one way. The whole work can be download at the following link:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Inder J. Taneja, Creative Magic Squares: Area Representations, Zenodo, June 22, pp. 1-45, 2021,&nbsp;<a rel=\"noreferrer noopener\" href=\"http:\/\/doi.org\/10.5281\/zenodo.5009224\" target=\"_blank\">http:\/\/doi.org\/10.5281\/zenodo.5009224<\/a><\/strong><\/li><li><strong>Inder J. Taneja, Creative Magic Squares: Area Representations With Fraction Numbers Entries (Version 2), Zenodo, August 16, 2021, 1-77,&nbsp;<a href=\"https:\/\/doi.org\/10.5281\/zenodo.5209502\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.5209502<\/a><\/strong><\/li><\/ul>\n\n\n\n<p>See below examples:<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-3\">Magic Squares of Order 3:<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/06\/3x3-block.png\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/06\/3x3-block.png\" alt=\"\" class=\"wp-image-8258\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-4\">Magic Squares of Order 4:<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/4x4-2x2-bord.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/4x4-2x2-bord.png?w=1024\" alt=\"\" class=\"wp-image-8858\"\/><\/a><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/06\/4x4-border.png\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/06\/4x4-border.png\" alt=\"\" class=\"wp-image-8264\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-4-fraction-numbers-entries\">Magic Squares of Order 4: Fraction Numbers Entries<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/4x4-2x2-frac-1.png?w=695\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/4x4-2x2-frac-1.png?w=695\" alt=\"\" class=\"wp-image-8791\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-5\">Magic Squares of Order 5<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/5x5-bord.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/5x5-bord.png?w=1024\" alt=\"\" class=\"wp-image-8850\"\/><\/a><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/06\/5x5-normal.png\" alt=\"\" class=\"wp-image-8269\"\/><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-6\">Magic Squares of Order 6<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/6x6-bord-1.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/6x6-bord-1.png?w=1024\" alt=\"\" class=\"wp-image-8842\"\/><\/a><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/06\/6x6-border.png\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/06\/6x6-border.png\" alt=\"\" class=\"wp-image-8274\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-6-fraction-numbers-entries\">Magic Squares of Order 6: Fraction Numbers Entries<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/6x6-2x2-bord-frac.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/6x6-2x2-bord-frac.png?w=1024\" alt=\"\" class=\"wp-image-8795\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-7\">Magic Squares of Order 7<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/7x7-bord-a.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/7x7-bord-a.png?w=1024\" alt=\"\" class=\"wp-image-8837\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-8\">Magic Squares of Order 8<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/8x8-2x2-2.png\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/8x8-2x2-2.png\" alt=\"\" class=\"wp-image-8828\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-8-fraction-numbers-entries\">Magic Squares of Order 8: Fraction Numbers Entries<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/8x8-4x4-2x2-frac.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/8x8-4x4-2x2-frac.png?w=1024\" alt=\"\" class=\"wp-image-8805\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-9\">Magic Squares of Order 9<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/9x9-3x3-1.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/9x9-3x3-1.png?w=1024\" alt=\"\" class=\"wp-image-8822\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-10\">Magic Squares of Order 10<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/8x8-bord.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/8x8-bord.png?w=1024\" alt=\"\" class=\"wp-image-8808\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-10-fraction-numbers-entries\">Magic Squares of Order 10: Fraction Numbers Entries<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/10x10-8x8-frac.png?w=1024\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/10x10-8x8-frac.png?w=1024\" alt=\"\" class=\"wp-image-8812\"\/><\/a><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"magic-squares-of-order-11\">Magic Squares of Order 11<\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/11x11-a.png\"><img decoding=\"async\" src=\"https:\/\/inderjtaneja.files.wordpress.com\/2021\/08\/11x11-a.png\" alt=\"\" class=\"wp-image-8817\"\/><\/a><\/figure><\/div>\n\n\n\n<p>More detailed number representations of each magic square are given in the following work for download:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Inder J. Taneja, Creative Magic Squares: Area Representations, Zenodo, June 22, pp. 1-45, 2021,\u00a0<a href=\"http:\/\/doi.org\/10.5281\/zenodo.5009224\" target=\"_blank\" rel=\"noreferrer noopener\">http:\/\/doi.org\/10.5281\/zenodo.5009224<\/a><\/strong><\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>It is well known that every magic square can be written as&nbsp;perfect square sum of entries. It is always possible with odd number entries starting from 1. In case of odd order magic squares we can also write with&nbsp;consecutive natural number&nbsp;entries. Still, it is unknown whether it is possible to even order magic squares. In [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":655,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3],"tags":[],"class_list":["post-654","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\/2022\/02\/4x4-bord-frac.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/654","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=654"}],"version-history":[{"count":1,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/654\/revisions"}],"predecessor-version":[{"id":656,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/654\/revisions\/656"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/655"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=654"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=654"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}