{"id":10336,"date":"2023-09-08T18:52:30","date_gmt":"2023-09-08T21:52:30","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=10336"},"modified":"2023-09-10T12:33:56","modified_gmt":"2023-09-10T15:33:56","slug":"different-types-of-magic-rectangles-in-construction-of-magic-squares-of-order-26","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=10336","title":{"rendered":"Different Types of Magic Rectangles in Construction of Magic Squares of Order 30"},"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 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 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:<\/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 30<\/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 30:<\/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=\"2295\" height=\"1163\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-1.png\" alt=\"\" class=\"wp-image-10337\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-1.png 2295w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-1-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-1-1024x519.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-1-768x389.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-1-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-1-1536x778.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-1-2048x1038.png 2048w\" sizes=\"(max-width: 2295px) 100vw, 2295px\" \/><\/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=\"2290\" height=\"1150\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-2.png\" alt=\"\" class=\"wp-image-10338\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-2.png 2290w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-2-300x151.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-2-1024x514.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-2-768x386.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-2-535x269.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-2-1536x771.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-2-2048x1028.png 2048w\" sizes=\"(max-width: 2290px) 100vw, 2290px\" \/><\/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 <br>Magic Squares of Order 30<\/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 <strong>magic squares of order 30<\/strong> centered in magic squares of order 6, 10, 14, 18 and 22 respectively. These are constructed with four equal sums <strong>bordered magic rectangles <\/strong>of orders <strong>12&#215;18, 10&#215;20, 8&#215;22, 6&#215;24<\/strong> and <strong>4&#215;26<\/strong> respectively.<\/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=\"2292\" height=\"1159\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-4.png\" alt=\"\" class=\"wp-image-10339\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-4.png 2292w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-4-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-4-1024x518.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-4-768x388.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-4-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-4-1536x777.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-4-2048x1036.png 2048w\" sizes=\"(max-width: 2292px) 100vw, 2292px\" \/><\/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=\"2283\" height=\"1158\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-30.png\" alt=\"\" class=\"wp-image-10340\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-30.png 2283w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-30-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-30-1024x519.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-30-768x390.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-30-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-30-1536x779.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-30-2048x1039.png 2048w\" sizes=\"(max-width: 2283px) 100vw, 2283px\" \/><\/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=\"2293\" height=\"1156\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-79.png\" alt=\"\" class=\"wp-image-10341\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-79.png 2293w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-79-300x151.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-79-1024x516.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-79-768x387.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-79-535x270.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-79-1536x774.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-79-2048x1032.png 2048w\" sizes=\"(max-width: 2293px) 100vw, 2293px\" \/><\/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=\"2319\" height=\"1161\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-151.png\" alt=\"\" class=\"wp-image-10342\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-151.png 2319w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-151-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-151-1024x513.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-151-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-151-535x268.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-151-1536x769.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-151-2048x1025.png 2048w\" sizes=\"(max-width: 2319px) 100vw, 2319px\" \/><\/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=\"2320\" height=\"1161\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-350.png\" alt=\"\" class=\"wp-image-10343\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-350.png 2320w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-350-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-350-1024x512.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-350-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-350-535x268.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-350-1536x769.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-350-2048x1025.png 2048w\" sizes=\"(max-width: 2320px) 100vw, 2320px\" \/><\/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=\"2319\" height=\"1161\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-401.png\" alt=\"\" class=\"wp-image-10344\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-401.png 2319w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-401-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-401-1024x513.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-401-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-401-535x268.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-401-1536x769.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-401-2048x1025.png 2048w\" sizes=\"(max-width: 2319px) 100vw, 2319px\" \/><\/figure>\n<\/div>\n\n\n<p>More examples are in <strong>pdf file <\/strong>attached at the end of work.<\/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%)\">2. Double Digits Magic Rectangles and <br>Magic Squares of Order 30<\/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 <strong>magic squares of order 30<\/strong> centered in magic squares of order 6, 10, 14 and 18 respectively. These are constructed with four equal sums <strong>double digits<\/strong> <strong>magic rectangles <\/strong>of orders <strong>12&#215;18, 10&#215;20, 8&#215;22<\/strong> and <strong>6&#215;24<\/strong> respectively.<\/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=\"2286\" height=\"1158\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-7.png\" alt=\"\" class=\"wp-image-10345\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-7.png 2286w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-7-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-7-1024x519.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-7-768x389.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-7-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-7-1536x778.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-7-2048x1037.png 2048w\" sizes=\"(max-width: 2286px) 100vw, 2286px\" \/><\/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=\"2291\" height=\"1164\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-36.png\" alt=\"\" class=\"wp-image-10346\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-36.png 2291w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-36-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-36-1024x520.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-36-768x390.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-36-535x272.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-36-1536x780.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-36-2048x1041.png 2048w\" sizes=\"(max-width: 2291px) 100vw, 2291px\" \/><\/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=\"2289\" height=\"1161\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-96.png\" alt=\"\" class=\"wp-image-10347\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-96.png 2289w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-96-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-96-1024x519.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-96-768x390.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-96-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-96-1536x779.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-96-2048x1039.png 2048w\" sizes=\"(max-width: 2289px) 100vw, 2289px\" \/><\/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=\"2316\" height=\"1160\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-194.png\" alt=\"\" class=\"wp-image-10348\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-194.png 2316w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-194-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-194-1024x513.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-194-768x385.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-194-535x268.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-194-1536x769.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-194-2048x1026.png 2048w\" sizes=\"(max-width: 2316px) 100vw, 2316px\" \/><\/figure>\n<\/div>\n\n\n<p>More examples are in <strong>pdf file <\/strong>attached at the end of work.<\/p>\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\">In this case we don&#8217;t have magic square of order 30 <strong>centered in magic square of order 22<\/strong> having four equal sums<strong> double digits magic rectangles<\/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(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles and <br>Magic Squares of Order 30: 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 <strong>magic squares of order 30<\/strong> centered in magic squares of order 6, 10, 14, 18 and 22 respectively. These are constructed with four equal sums <strong>cornered magic rectangles <\/strong>of orders <strong>12&#215;18, 10&#215;20, 8&#215;22, 6&#215;24<\/strong> and <strong>4&#215;26<\/strong> respectively.<\/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=\"2290\" height=\"1157\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-14.png\" alt=\"\" class=\"wp-image-10349\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-14.png 2290w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-14-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-14-1024x517.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-14-768x388.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-14-535x270.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-14-1536x776.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-14-2048x1035.png 2048w\" sizes=\"(max-width: 2290px) 100vw, 2290px\" \/><\/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=\"2287\" height=\"1156\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-45.png\" alt=\"\" class=\"wp-image-10352\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-45.png 2287w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-45-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-45-1024x518.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-45-768x388.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-45-535x270.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-45-1536x776.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-45-2048x1035.png 2048w\" sizes=\"(max-width: 2287px) 100vw, 2287px\" \/><\/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=\"2320\" height=\"1159\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-112.png\" alt=\"\" class=\"wp-image-10355\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-112.png 2320w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-112-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-112-1024x512.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-112-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-112-535x267.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-112-1536x767.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-112-2048x1023.png 2048w\" sizes=\"(max-width: 2320px) 100vw, 2320px\" \/><\/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=\"2322\" height=\"1159\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-229.png\" alt=\"\" class=\"wp-image-10358\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-229.png 2322w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-229-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-229-1024x511.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-229-768x383.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-229-535x267.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-229-1536x767.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-229-2048x1022.png 2048w\" sizes=\"(max-width: 2322px) 100vw, 2322px\" \/><\/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=\"2320\" height=\"1161\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-458.png\" alt=\"\" class=\"wp-image-10361\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-458.png 2320w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-458-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-458-1024x512.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-458-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-458-535x268.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-458-1536x769.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-458-2048x1025.png 2048w\" sizes=\"(max-width: 2320px) 100vw, 2320px\" \/><\/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 26: 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 are examples of <strong>magic squares of order 30<\/strong> centered in magic squares of order 6, 10, 14 and 18 respectively. These are constructed with four equal sums <strong>cornered magic rectangles <\/strong>of orders <strong>12&#215;18, 10&#215;20, 8&#215;22<\/strong> and <strong>6&#215;24<\/strong> respectively.<\/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=\"2293\" height=\"1162\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-15.png\" alt=\"\" class=\"wp-image-10350\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-15.png 2293w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-15-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-15-1024x519.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-15-768x389.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-15-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-15-1536x778.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-15-2048x1038.png 2048w\" sizes=\"(max-width: 2293px) 100vw, 2293px\" \/><\/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=\"2290\" height=\"1159\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-60.png\" alt=\"\" class=\"wp-image-10353\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-60.png 2290w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-60-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-60-1024x518.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-60-768x389.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-60-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-60-1536x777.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-60-2048x1037.png 2048w\" sizes=\"(max-width: 2290px) 100vw, 2290px\" \/><\/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=\"2323\" height=\"1158\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-129.png\" alt=\"\" class=\"wp-image-10356\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-129.png 2323w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-129-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-129-1024x510.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-129-768x383.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-129-535x267.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-129-1536x766.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-129-2048x1021.png 2048w\" sizes=\"(max-width: 2323px) 100vw, 2323px\" \/><\/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=\"2325\" height=\"1163\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-283.png\" alt=\"\" class=\"wp-image-10359\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-283.png 2325w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-283-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-283-1024x512.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-283-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-283-535x268.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-283-1536x768.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-283-2048x1024.png 2048w\" sizes=\"(max-width: 2325px) 100vw, 2325px\" \/><\/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 30: 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 <strong>magic squares of order 30<\/strong> centered in magic squares of order 6, 10, 14 and 18 respectively. These are constructed with four equal sums <strong>cornered magic rectangles <\/strong>of orders <strong>12&#215;18, 10&#215;20, 8&#215;22<\/strong> and <strong>6&#215;24<\/strong> respectively.<\/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=\"2293\" height=\"1160\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-19.png\" alt=\"\" class=\"wp-image-10351\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-19.png 2293w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-19-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-19-1024x518.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-19-768x389.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-19-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-19-1536x777.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-19-2048x1036.png 2048w\" sizes=\"(max-width: 2293px) 100vw, 2293px\" \/><\/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=\"2292\" height=\"1162\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-67.png\" alt=\"\" class=\"wp-image-10354\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-67.png 2292w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-67-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-67-1024x519.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-67-768x389.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-67-535x271.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-67-1536x779.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-67-2048x1038.png 2048w\" sizes=\"(max-width: 2292px) 100vw, 2292px\" \/><\/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=\"2318\" height=\"1158\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-144.png\" alt=\"\" class=\"wp-image-10357\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-144.png 2318w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-144-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-144-1024x512.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-144-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-144-535x267.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-144-1536x767.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-144-2048x1023.png 2048w\" sizes=\"(max-width: 2318px) 100vw, 2318px\" \/><\/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=\"2322\" height=\"1157\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-326.png\" alt=\"\" class=\"wp-image-10360\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-326.png 2322w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-326-300x149.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-326-1024x510.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-326-768x383.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-326-535x267.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-326-1536x765.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30-326-2048x1020.png 2048w\" sizes=\"(max-width: 2322px) 100vw, 2322px\" \/><\/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 30<\/h3>\n\n\n\n<p class=\"has-text-align-justify\">Below is a pdf file of 541 magic squares of order 30 for download.  These are constructed with 4 <strong>equal sums magic rectangles of three types<\/strong> as explained above. Magic squares numbers 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\/30x30.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of 30x30.\"><\/object><a id=\"wp-block-file--media-60b0ed1b-a33d-4154-ab2b-b7014de0b610\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30.pdf\">30&#215;30<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/30x30.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<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":10358,"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-10336","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\/30x30-229.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10336","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=10336"}],"version-history":[{"count":3,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10336\/revisions"}],"predecessor-version":[{"id":10415,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10336\/revisions\/10415"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/10358"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10336"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10336"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}