{"id":10047,"date":"2023-08-11T13:17:11","date_gmt":"2023-08-11T16:17:11","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=10047"},"modified":"2023-09-10T10:19:39","modified_gmt":"2023-09-10T13:19:39","slug":"cornered-magic-rectangles-in-construction-of-magic-squares-magic-squares-of-order-18","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=10047","title":{"rendered":"Different Types of Magic Rectangles in Construction of Magic Squares of Orders 14 and 18"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><strong><em><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">Never Seen Before<\/mark><\/em><\/strong><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-pale-ocean-gradient-background has-background\">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>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. 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>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<h4 class=\"wp-block-heading\">Example 2. Bordered Magic Rectangle of Order 6&#215;16<\/h4>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"1024\" height=\"231\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-6x16-2.png\" alt=\"\" class=\"wp-image-10226\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-6x16-2.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-6x16-2-300x68.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-6x16-2-768x173.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-6x16-2-535x121.png 535w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<p>The entries according to colors are as follows:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1066\" height=\"286\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-2.png\" alt=\"\" class=\"wp-image-10227\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-2.png 1066w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-2-300x80.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-2-1024x275.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-2-768x206.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-2-535x144.png 535w\" sizes=\"(max-width: 1066px) 100vw, 1066px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p><\/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><\/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>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 loading=\"lazy\" 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><\/p>\n\n\n\n<p>Except the corners, the entries <strong>entries sums<\/strong> are understood as<\/p>\n\n\n\n<p><\/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><\/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><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-left\"><strong>Example 2. Double Digits Magic Rectangle of Order 12&#215;18<\/strong><\/h4>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1449\" height=\"421\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-12x22-2.png\" alt=\"\" class=\"wp-image-10235\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-12x22-2.png 1449w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-12x22-2-300x87.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-12x22-2-1024x298.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-12x22-2-768x223.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-12x22-2-535x155.png 535w\" sizes=\"(max-width: 1449px) 100vw, 1449px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>Except the corners, the entries <strong>entries sums<\/strong> are understood as<\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1153\" height=\"493\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-3-1.png\" alt=\"\" class=\"wp-image-10236\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-3-1.png 1153w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-3-1-300x128.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-3-1-1024x438.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-3-1-768x328.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-3-1-535x229.png 535w\" sizes=\"(max-width: 1153px) 100vw, 1153px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"941\" height=\"268\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-4-1.png\" alt=\"\" class=\"wp-image-10237\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-4-1.png 941w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-4-1-300x85.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-4-1-768x219.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-4-1-535x152.png 535w\" sizes=\"(max-width: 941px) 100vw, 941px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"951\" height=\"166\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-4x14-1.png\" alt=\"\" class=\"wp-image-10239\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-4x14-1.png 951w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-4x14-1-300x52.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-4x14-1-768x134.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-4x14-1-535x93.png 535w\" sizes=\"(max-width: 951px) 100vw, 951px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify\">After reorganizing the <strong>internal magic rectangle of order 4&#215;16<\/strong>, we have the following <strong>double digits magic rectangle<\/strong>:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1544\" height=\"462\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DDB-12x22-1.png\" alt=\"\" class=\"wp-image-10238\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DDB-12x22-1.png 1544w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DDB-12x22-1-300x90.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DDB-12x22-1-1024x306.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DDB-12x22-1-768x230.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DDB-12x22-1-535x160.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DDB-12x22-1-1536x460.png 1536w\" sizes=\"(max-width: 1544px) 100vw, 1544px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p><\/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><\/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>Below are two examples of cornered magic rectangles<\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-left\"><strong>Example 1. Cornered Magic Rectangle of Order 8&#215;12<\/strong><\/h4>\n\n\n\n<p><\/p>\n\n\n\n<p>Let&#8217;s consider a cornered magic rectangle of order <strong>8&#215;12<\/strong> formed by 96 sequencial entries, i.e., from 1 to 96:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"806\" height=\"323\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-8x12-1.png\" alt=\"\" class=\"wp-image-10083\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-8x12-1.png 806w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-8x12-1-300x120.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-8x12-1-768x308.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-8x12-1-535x214.png 535w\" sizes=\"(max-width: 806px) 100vw, 806px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>Distributions in colors as follows:<\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2240\" height=\"257\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/entries-1.png\" alt=\"\" class=\"wp-image-10113\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/entries-1.png 2240w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/entries-1-300x34.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/entries-1-1024x117.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/entries-1-768x88.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/entries-1-535x61.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/entries-1-1536x176.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/entries-1-2048x235.png 2048w\" sizes=\"(max-width: 2240px) 100vw, 2240px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-left\"><strong>Example 2. Cornered Magic Rectangle of Order 8&#215;12<\/strong><\/h4>\n\n\n\n<p><\/p>\n\n\n\n<p>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<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><\/p>\n\n\n\n<p>Distributions in colors as follows:<\/p>\n\n\n\n<p><\/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><\/p>\n\n\n\n<p>More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p><\/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><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 42%,rgb(255,105,0) 100%)\">Based on ideas given above, we shall construct magic squares of order 14 and 18. These are constructed with <strong>four magic rectangles<\/strong>. As explained above these are of are of <strong>equal sums in pairs except in corners.<\/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 never seen in the history. There are of different styles of this magic squares of order 14. See below:<\/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(255,203,112) 0%,rgb(82,199,95) 50%,rgb(65,88,208) 100%)\">Magic Squares of Order 14<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p>Initially below are two <strong>double digits<\/strong> and <strong>cornered magic squares<\/strong> of order 14:<\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2110\" height=\"567\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-1-2.png\" alt=\"\" class=\"wp-image-10087\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-1-2.png 2110w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-1-2-300x81.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-1-2-1024x275.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-1-2-768x206.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-1-2-535x144.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-1-2-1536x413.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-1-2-2048x550.png 2048w\" sizes=\"(max-width: 2110px) 100vw, 2110px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">1. Bordered Magic Rectangles and Magic Squares of Order 14<\/h3>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2114\" height=\"567\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2a.png\" alt=\"\" class=\"wp-image-10210\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2a.png 2114w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2a-300x80.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2a-1024x275.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2a-768x206.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2a-535x143.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2a-1536x412.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2a-2048x549.png 2048w\" sizes=\"(max-width: 2114px) 100vw, 2114px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2114\" height=\"566\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2b.png\" alt=\"\" class=\"wp-image-10211\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2b.png 2114w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2b-300x80.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2b-1024x274.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2b-768x206.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2b-535x143.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2b-1536x411.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-14x14-1-2b-2048x548.png 2048w\" sizes=\"(max-width: 2114px) 100vw, 2114px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">2. Cornered Magic Rectangles and Magic Squares of Order 14: First Type<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2113\" height=\"566\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-3-4.png\" alt=\"\" class=\"wp-image-10088\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-3-4.png 2113w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-3-4-300x80.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-3-4-1024x274.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-3-4-768x206.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-3-4-535x143.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-3-4-1536x411.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-3-4-2048x549.png 2048w\" sizes=\"(max-width: 2113px) 100vw, 2113px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2113\" height=\"559\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-5-6-2.png\" alt=\"\" class=\"wp-image-10104\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-5-6-2.png 2113w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-5-6-2-300x79.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-5-6-2-1024x271.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-5-6-2-768x203.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-5-6-2-535x142.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-5-6-2-1536x406.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-5-6-2-2048x542.png 2048w\" sizes=\"(max-width: 2113px) 100vw, 2113px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">2. Cornered Magic Rectangles and Magic Squares of Order 14: Second Type<\/h3>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2650\" height=\"706\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-18x18-7-8.png\" alt=\"\" class=\"wp-image-10091\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-18x18-7-8.png 2650w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-18x18-7-8-300x80.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-18x18-7-8-1024x273.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-18x18-7-8-768x205.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-18x18-7-8-535x143.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-18x18-7-8-1536x409.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-18x18-7-8-2048x546.png 2048w\" sizes=\"(max-width: 2650px) 100vw, 2650px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2116\" height=\"564\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-9-10.png\" alt=\"\" class=\"wp-image-10092\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-9-10.png 2116w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-9-10-300x80.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-9-10-1024x273.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-9-10-768x205.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-9-10-535x143.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-9-10-1536x409.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/MR-14x14-9-10-2048x546.png 2048w\" sizes=\"(max-width: 2116px) 100vw, 2116px\" \/><\/figure>\n<\/div>\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(255,203,112) 0%,rgb(82,199,95) 50%,rgb(65,88,208) 100%)\">Magic Squares of Order 18<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">Initially below are two <strong>double digits<\/strong> and <strong>cornered magic squares<\/strong> of order 18:<\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1316\" height=\"704\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-1.png\" alt=\"\" class=\"wp-image-10247\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-1.png 1316w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-1-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-1-1024x548.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-1-768x411.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-1-535x286.png 535w\" sizes=\"(max-width: 1316px) 100vw, 1316px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1318\" height=\"703\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-2.png\" alt=\"\" class=\"wp-image-10248\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-2.png 1318w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-2-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-2-1024x546.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-2-768x410.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-2-535x285.png 535w\" sizes=\"(max-width: 1318px) 100vw, 1318px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">1. Bordered Magic Rectangles and Magic Squares of Order 18<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">Below are examples of magic <strong>squares of order 18<\/strong> centered in magic squares of order 6 and 10 respectively. These are constructed with four equal sums <strong>bordered magic rectangles <\/strong>of orders <strong>6&#215;12<\/strong> and <strong>4&#215;14<\/strong><\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1314\" height=\"709\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-3.png\" alt=\"\" class=\"wp-image-10246\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-3.png 1314w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-3-300x162.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-3-1024x553.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-3-768x414.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-3-535x289.png 535w\" sizes=\"(max-width: 1314px) 100vw, 1314px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1313\" height=\"704\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-29.png\" alt=\"\" class=\"wp-image-10256\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-29.png 1313w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-29-300x161.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-29-1024x549.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-29-768x412.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-29-535x287.png 535w\" sizes=\"(max-width: 1313px) 100vw, 1313px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">2. Double Digits Magic Rectangles and Magic Squares of Order 18<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">Below is a <strong>magic square of order 18<\/strong> centered in magic squares of order 6. It is constructed with four equal sums <strong>double digits magic rectangles <\/strong>of order <strong>6&#215;12.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1311\" height=\"710\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-7.png\" alt=\"\" class=\"wp-image-10249\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-7.png 1311w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-7-300x162.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-7-1024x555.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-7-768x416.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-7-535x290.png 535w\" sizes=\"(max-width: 1311px) 100vw, 1311px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p class=\"has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">In this case we don&#8217;t have magic square of order 18 centered in magic square of order 10 having four equal sums double digits magic rectangles.<\/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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles and <br>Magic Squares of Order 18: First Type<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">Below are examples of magic <strong>squares of order 18<\/strong> centered in magic squares of order 6 and 10 respectively. These are constructed with four equal sums <strong>cornered magic rectangles <\/strong>of orders <strong>6&#215;12<\/strong> and <strong>4&#215;14<\/strong><\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1312\" height=\"706\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-14.png\" alt=\"\" class=\"wp-image-10250\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-14.png 1312w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-14-300x161.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-14-1024x551.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-14-768x413.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-14-535x288.png 535w\" sizes=\"(max-width: 1312px) 100vw, 1312px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1315\" height=\"711\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-36.png\" alt=\"\" class=\"wp-image-10254\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-36.png 1315w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-36-300x162.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-36-1024x554.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-36-768x415.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-36-535x289.png 535w\" sizes=\"(max-width: 1315px) 100vw, 1315px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles and <br>Magic Squares of Order 18: Second Type<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">Below is an example of <strong>magic square of order 18<\/strong> centered in magic squares of order 6. It is constructed with four equal sums <strong>cornered magic rectangles <\/strong>of orders <strong>6&#215;12<\/strong><\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1308\" height=\"709\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-15-1.png\" alt=\"\" class=\"wp-image-10252\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-15-1.png 1308w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-15-1-300x163.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-15-1-1024x555.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-15-1-768x416.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-15-1-535x290.png 535w\" sizes=\"(max-width: 1308px) 100vw, 1308px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles and <br>Magic Squares of Order 18: Third Type<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">Below are examples of magic <strong>squares of order 18<\/strong> centered in magic squares of order 6 and 10 respectively. These are constructed with four equal sums <strong>cornered magic rectangles <\/strong>of orders <strong>6&#215;12<\/strong> and <strong>4&#215;14<\/strong><\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1315\" height=\"706\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-22.png\" alt=\"\" class=\"wp-image-10253\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-22.png 1315w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-22-300x161.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-22-1024x550.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-22-768x412.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-22-535x287.png 535w\" sizes=\"(max-width: 1315px) 100vw, 1315px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1315\" height=\"706\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-53-1.png\" alt=\"\" class=\"wp-image-10261\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-53-1.png 1315w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-53-1-300x161.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-53-1-1024x550.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-53-1-768x412.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-18x18-53-1-535x287.png 535w\" sizes=\"(max-width: 1315px) 100vw, 1315px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">PDF File of Magic Squares of Order 18<\/h3>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">Below is a pdf file of 61 magic squares of order 18 for download.  These are constructed with 4 <strong>equal sums magic rectangles of three types<\/strong> as explained above. Magic squares number 1 and 2 are basic.<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/18x18.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of 18x18.\"><\/object><a id=\"wp-block-file--media-60b0ed1b-a33d-4154-ab2b-b7014de0b610\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/18x18.pdf\">18&#215;18<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/18x18.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-60b0ed1b-a33d-4154-ab2b-b7014de0b610\">Download<\/a><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center has-blush-light-purple-gradient-background has-background\">References<\/h2>\n\n\n\n<p><\/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>, <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, <strong>Zenodo<\/strong>. <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8230214\">https:\/\/doi.org\/10.5281\/zenodo.8230214<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, New Concepts in Magic Squares: Cornered Magic Squares of Orders 5 to 81, pp. 1-27, August 09, 2023,  <strong>Zenodo<\/strong>. <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8231157\">https:\/\/doi.org\/10.5281\/zenodo.8231157<\/a>.<\/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>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, 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>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, 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>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, 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>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, 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>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, 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>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, 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><\/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":10259,"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-10047","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\/09\/MR-18x18-36-1.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10047","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=10047"}],"version-history":[{"count":21,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10047\/revisions"}],"predecessor-version":[{"id":10408,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10047\/revisions\/10408"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/10259"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10047"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10047"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10047"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}