{"id":10199,"date":"2023-08-23T20:51:20","date_gmt":"2023-08-23T23:51:20","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=10199"},"modified":"2023-09-10T12:55:36","modified_gmt":"2023-09-10T15:55:36","slug":"new-concepts-in-magic-squares-cornered-magic-rectangles-in-construction-of-even-orders-magic-squares","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=10199","title":{"rendered":"Different Types of Magic Rectangles in Construction of Even Orders Magic Squares"},"content":{"rendered":"\n<p class=\"has-text-align-right wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\"><strong><em>Never Seen Before<\/em><\/strong><\/mark>.<\/p>\n\n\n\n<p class=\"has-text-align-justify has-pale-ocean-gradient-background has-background wp-block-paragraph\">This work brings magic squares in very different way. These are based on three different types of <strong>magic rectangles<\/strong>:<br><strong>1. Bordered Magic Rectangles:<br>2. Double Digits Magic Rectangles:<br>3. Cornered magic rectangles. <br><\/strong><br>First of let&#8217;s understand these magic rectangles one by one.  Removing external borders in each case still we left with lower order <strong>magic rectangles<\/strong>. See below few examples in each case.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">1. Bordered Magic Rectangles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Below are two examples of bordered magic rectangles<\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-left\"><strong>Example. Bordered Magic Rectangle of Order 12&#215;18<\/strong><\/h4>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"1141\" height=\"420\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-12x18-1.png\" alt=\"\" class=\"wp-image-10220\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-12x18-1.png 1141w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-12x18-1-300x110.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-12x18-1-1024x377.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-12x18-1-768x283.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-12x18-1-535x197.png 535w\" sizes=\"(max-width: 1141px) 100vw, 1141px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The entries according to colors are as follows:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"1491\" height=\"606\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1.png\" alt=\"\" class=\"wp-image-10221\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1.png 1491w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1-300x122.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1-1024x416.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1-768x312.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1-535x217.png 535w\" sizes=\"(max-width: 1491px) 100vw, 1491px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Inder J. Taneja, <\/strong>Different Types of Magic Rectangles, <strong>Zenodo<\/strong>, September 04, 2023, pp. 1-26, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8316719\">https:\/\/doi.org\/10.5281\/zenodo.8316719<\/a>.<\/mark><\/li>\n<\/ul>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">2. Double Digits Magic Rectangles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Below are two examples of bordered magic rectangles<\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-left\"><strong>Example 1. Double Digits Magic Rectangle of Order 14&#215;20<\/strong><\/h4>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"1270\" height=\"490\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-14x20-1.png\" alt=\"\" class=\"wp-image-10232\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-14x20-1.png 1270w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-14x20-1-300x116.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-14x20-1-1024x395.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-14x20-1-768x296.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-14x20-1-535x206.png 535w\" sizes=\"(max-width: 1270px) 100vw, 1270px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Except the corners, the entries <strong>entries sums<\/strong> are understood as<\/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 loading=\"lazy\" decoding=\"async\" width=\"1022\" height=\"879\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1.png\" alt=\"\" class=\"wp-image-10233\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1.png 1022w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1-300x258.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1-768x661.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1-535x460.png 535w\" sizes=\"(max-width: 1022px) 100vw, 1022px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"949\" height=\"336\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2.png\" alt=\"\" class=\"wp-image-10234\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2.png 949w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2-300x106.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2-768x272.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2-535x189.png 535w\" sizes=\"(max-width: 949px) 100vw, 949px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Inder J. Taneja, <\/strong>Different Types of Magic Rectangles, <strong>Zenodo<\/strong>, September 04, 2023, pp. 1-26, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8316719\">https:\/\/doi.org\/10.5281\/zenodo.8316719<\/a><\/mark><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">.<\/mark><\/li>\n<\/ul>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Below is an example of <strong>cornered magic rectangles<\/strong><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-left\"><strong>Example. Cornered Magic Rectangle of Order 8&#215;12<\/strong><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s consider a cornered magic rectangle of order <strong>10&#215;24<\/strong> formed by 240 sequencial entries, i.e., from 1 to 240:<\/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 loading=\"lazy\" decoding=\"async\" width=\"1680\" height=\"394\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1.png\" alt=\"\" class=\"wp-image-10244\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1.png 1680w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-300x70.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-1024x240.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-768x180.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-535x125.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-1536x360.png 1536w\" sizes=\"(max-width: 1680px) 100vw, 1680px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Distributions in colors is as follows:<\/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 loading=\"lazy\" decoding=\"async\" width=\"1707\" height=\"501\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2.png\" alt=\"\" class=\"wp-image-10245\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2.png 1707w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-300x88.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-1024x301.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-768x225.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-535x157.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-1536x451.png 1536w\" sizes=\"(max-width: 1707px) 100vw, 1707px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Inder J. Taneja, <\/strong>Different Types of Magic Rectangles, <strong>Zenodo<\/strong>, September 04, 2023, pp. 1-26, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8316719\">https:\/\/doi.org\/10.5281\/zenodo.8316719<\/a><\/mark><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">.<\/mark><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-paragraph\">Based on ideas given above, we shall construct magic squares of order 22. These are constructed with <strong>four magic rectangles<\/strong>. As explained above these are of are of <strong>equal sums<\/strong>. These are made with the help of script by H. White (<a href=\"https:\/\/budshaw.ca\/Download.html\">Downloads (budshaw.ca<\/a>) &#8211; <em>NestedCornerRectangles<\/em>)). These kinds of magic squares are <strong>never seen in the history<\/strong>. There are of different styles. All the three types of magic rectangles are used to bring magic squares of order 22. See below the summary.<\/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(2,3,129) 0%,rgb(255,255,255) 33%,rgb(40,116,252) 100%)\">SUMMARY: Even Orders Magic Squares from Orders 14 to 30:<br><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">Bordered, Double Digits and Cornered Magic Rectangles<\/mark><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ol class=\"has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-list\">\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=9406\" target=\"_blank\">Double Digits Even and Odd Orders Bordered Magic Squares<\/a>.<br>Also see: New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108, pp. 1-30, August 09, 2023,&nbsp;<strong>Zenodo<\/strong>.&nbsp;<a href=\"https:\/\/doi.org\/10.5281\/zenodo.8230214\">https:\/\/doi.org\/10.5281\/zenodo.8230214<\/a>.<br><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=9400\">Cornered Magic Squares<\/a>.<br>Also see: New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 81, pp. 1-27, August 09, 2023,&nbsp;<strong>Zenodo<\/strong>.&nbsp;<a href=\"https:\/\/doi.org\/10.5281\/zenodo.8231157\">https:\/\/doi.org\/10.5281\/zenodo.8231157<\/a>.<br><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Different Types of Magic Rectangles, <strong>Zenodo<\/strong>, September 04, 2023, pp. 1-26, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8316719\">https:\/\/doi.org\/10.5281\/zenodo.8316719<\/a>.<br><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,\u00a0<a href=\"https:\/\/numbers-magic.com\/?p=10047\" data-type=\"link\" data-id=\"https:\/\/numbers-magic.com\/?p=10047\">Different Types of Magic Rectangles in Construction of Magic Squares of Orders 14 and 18<\/a>.<br>Also see: Different Types of Magic Rectangles in Construction of Magic Squares of Orders 14 and 18, <strong>Zenodo<\/strong>, September 10, 2023, pp. 1-32, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8331709\">https:\/\/doi.org\/10.5281\/zenodo.8331709<\/a>.<br><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=10061\">Different Types of Magic Rectangles in Construction of Magic Squares of Order 22.<\/a><br>Also see: Different Types of Magic Rectangles in Construction of Magic Squares of Order 22, <strong>Zenodo<\/strong>, September 10, 2023, pp. 1-36, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8331743\">https:\/\/doi.org\/10.5281\/zenodo.8331743<\/a>.<br><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,<a href=\"https:\/\/numbers-magic.com\/?p=10071\"> Different Types of Magic Rectangles in Construction of Magic Squares of Order 26<\/a>.<br>Also see: Different Types of Magic Rectangles in Construction of Magic Squares of Order 26, <strong>Zenodo<\/strong>, September 10, 2023, pp. 1-39, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8331750\">https:\/\/doi.org\/10.5281\/zenodo.8331750<\/a>.<br><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, <a href=\"https:\/\/numbers-magic.com\/?p=10071\"><\/a><a href=\"https:\/\/numbers-magic.com\/?p=10336\">Different Types of Magic Rectangles in Construction of Magic Squares of Order 30.<\/a> Also see: Different Types of Magic Rectangles in Construction of Magic Squares of Order 30, <strong>Zenodo<\/strong>, September 10, 2023, pp. 1-44, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8331755\">https:\/\/doi.org\/10.5281\/zenodo.8331755<\/a>.<br><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, <a href=\"https:\/\/numbers-magic.com\/?p=10365\">Different Types of Magic Rectangles in Construction of Magic Squares of Order 34.<\/a> Also see: Different Types of Magic Rectangles in Construction of Magic Squares of Order 34, <strong>Zenodo<\/strong>, September 10, 2023, pp. 1-49, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8331759\">https:\/\/doi.org\/10.5281\/zenodo.8331759<\/a>.<br><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, <a href=\"https:\/\/numbers-magic.com\/?p=10172\">Cornered Magic Squares in Construction of Magic Squares of Orders 16, 20, 24 and 28<\/a><br>Also see: Cornered Magic Squares in Construction of Magic Squares of Orders 16, 20, 24 and 28, <strong>Zenodo<\/strong>, September 10, 2023, pp. 1-35, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8332156\">https:\/\/doi.org\/10.5281\/zenodo.8332156<\/a>.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Never Seen Before. This work brings magic squares in very different way. These are based on three different types of magic rectangles:1. Bordered Magic Rectangles:2. Double Digits Magic Rectangles:3. Cornered magic rectangles. First of let&#8217;s understand these magic rectangles one by one. Removing external borders in each case still we left with lower order magic [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":10154,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[39],"tags":[],"class_list":["post-10199","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-summary"],"jetpack_featured_media_url":"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-26x26-3-1.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10199","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=10199"}],"version-history":[{"count":12,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10199\/revisions"}],"predecessor-version":[{"id":11780,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10199\/revisions\/11780"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/10154"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10199"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10199"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10199"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}