{"id":10061,"date":"2023-08-11T20:24:01","date_gmt":"2023-08-11T23:24:01","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=10061"},"modified":"2023-09-10T10:21:09","modified_gmt":"2023-09-10T13:21:09","slug":"cornered-magic-rectangles-in-construction-of-magic-squares-magic-squares-of-order-22","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=10061","title":{"rendered":"Different Types of Magic Rectangles in Construction of Magic Squares of Order 22"},"content":{"rendered":"\n<p class=\"has-text-align-right wp-block-paragraph\"><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 wp-block-paragraph\">This work brings magic squares in very different way. These are based on three different types of <strong>magic rectangles<\/strong>:<br><strong>1. Bordered Magic Rectangles:<br>2. Double Digits Magic Rectangles:<br>3. Cornered magic rectangles. <br><\/strong><br>First of let&#8217;s understand these magic rectangles one by one.  Removing external borders in each case still we left with lower order <strong>magic rectangles<\/strong>. See below few examples in each case.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">1. Bordered Magic Rectangles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Below are two examples of bordered magic rectangles<\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-left\"><strong>Example 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 class=\"wp-block-paragraph\">The entries according to colors are as follows:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"1491\" height=\"606\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1.png\" alt=\"\" class=\"wp-image-10221\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1.png 1491w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1-300x122.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1-1024x416.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1-768x312.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/BMR-entries-1-535x217.png 535w\" sizes=\"(max-width: 1491px) 100vw, 1491px\" \/><\/figure>\n<\/div>\n\n\n<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 class=\"wp-block-paragraph\">The entries according to colors are as follows:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img 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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Inder J. Taneja, <\/strong>Different Types of Magic Rectangles, <strong>Zenodo<\/strong>, September 04, 2023, pp. 1-26, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8316719\">https:\/\/doi.org\/10.5281\/zenodo.8316719<\/a>.<\/mark><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">2. Double Digits Magic Rectangles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Below are two examples of bordered magic rectangles<\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-left\"><strong>Example 1. Double Digits Magic Rectangle of Order 14&#215;20<\/strong><\/h4>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img 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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Except the corners, the entries <strong>entries sums<\/strong> are understood as<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1022\" height=\"879\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1.png\" alt=\"\" class=\"wp-image-10233\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1.png 1022w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1-300x258.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1-768x661.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-1-1-535x460.png 535w\" sizes=\"(max-width: 1022px) 100vw, 1022px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"949\" height=\"336\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2.png\" alt=\"\" class=\"wp-image-10234\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2.png 949w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2-300x106.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2-768x272.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/DD-sums-2-535x189.png 535w\" sizes=\"(max-width: 949px) 100vw, 949px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Except the corners, the entries <strong>entries sums<\/strong> are understood as<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"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 class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"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 class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Inder J. Taneja, <\/strong>Different Types of Magic Rectangles, <strong>Zenodo<\/strong>, September 04, 2023, pp. 1-26, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8316719\">https:\/\/doi.org\/10.5281\/zenodo.8316719<\/a><\/mark><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">.<\/mark><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Below 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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s consider a cornered magic rectangle of order <strong>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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Distributions in colors as follows:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"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 class=\"wp-block-paragraph\"><\/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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let&#8217;s consider a cornered magic rectangle of order <strong>10&#215;24<\/strong> formed by 240 sequencial entries, i.e., from 1 to 240:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1680\" height=\"394\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1.png\" alt=\"\" class=\"wp-image-10244\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1.png 1680w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-300x70.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-1024x240.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-768x180.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-535x125.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/CMR-10x24-1-1536x360.png 1536w\" sizes=\"(max-width: 1680px) 100vw, 1680px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Distributions in colors as follows:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1707\" height=\"501\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2.png\" alt=\"\" class=\"wp-image-10245\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2.png 1707w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-300x88.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-1024x301.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-768x225.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-535x157.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/entries-2-1536x451.png 1536w\" sizes=\"(max-width: 1707px) 100vw, 1707px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">More details are given in author&#8217;s recent work:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Inder J. Taneja, <\/strong>Different Types of Magic Rectangles, <strong>Zenodo<\/strong>, September 04, 2023, pp. 1-26, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8316719\">https:\/\/doi.org\/10.5281\/zenodo.8316719<\/a><\/mark><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">.<\/mark><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" 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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(255,203,112) 0%,rgb(82,199,95) 50%,rgb(65,88,208) 100%)\">Magic Squares of Order 22<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Initially below are two <strong>double digits<\/strong> and <strong>cornered magic squares<\/strong> of order 22:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1596\" height=\"835\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-1.png\" alt=\"\" class=\"wp-image-10267\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-1.png 1596w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-1-300x157.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-1-1024x536.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-1-768x402.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-1-535x280.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-1-1536x804.png 1536w\" sizes=\"(max-width: 1596px) 100vw, 1596px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1598\" height=\"841\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-2.png\" alt=\"\" class=\"wp-image-10268\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-2.png 1598w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-2-300x158.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-2-1024x539.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-2-768x404.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-2-535x282.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-2-1536x808.png 1536w\" sizes=\"(max-width: 1598px) 100vw, 1598px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">1. Bordered Magic Rectangles and <br>Magic Squares of Order 22<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-paragraph\">Below are examples of <strong>magic squares of order 22<\/strong> centered in magic squares of order 6, 10 and 14 respectively. These are constructed with four equal sums <strong>bordered magic rectangles <\/strong>of orders <strong>8&#215;14, 6&#215;16<\/strong> and <strong>4&#215;18<\/strong> respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1596\" height=\"852\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-4.png\" alt=\"\" class=\"wp-image-10269\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-4.png 1596w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-4-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-4-1024x547.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-4-768x410.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-4-535x286.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-4-1536x820.png 1536w\" sizes=\"(max-width: 1596px) 100vw, 1596px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1597\" height=\"847\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-30.png\" alt=\"\" class=\"wp-image-10270\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-30.png 1597w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-30-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-30-1024x543.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-30-768x407.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-30-535x284.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-30-1536x815.png 1536w\" sizes=\"(max-width: 1597px) 100vw, 1597px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1603\" height=\"842\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-89.png\" alt=\"\" class=\"wp-image-10271\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-89.png 1603w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-89-300x158.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-89-1024x538.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-89-768x403.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-89-535x281.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-89-1536x807.png 1536w\" sizes=\"(max-width: 1603px) 100vw, 1603px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">More examples are in <strong>pdf file <\/strong>attached at the end of work.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(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 22<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-paragraph\">Below are examples of <strong>magic squares of order 22<\/strong> centered in magic squares of order 6 and 10 respectively. These are constructed with four equal sums <strong>double digits magic rectangles <\/strong>of orders <strong>8&#215;14 <\/strong>and<strong> 6&#215;16<\/strong> respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1596\" height=\"846\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-7-1.png\" alt=\"\" class=\"wp-image-10273\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-7-1.png 1596w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-7-1-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-7-1-1024x543.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-7-1-768x407.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-7-1-535x284.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-7-1-1536x814.png 1536w\" sizes=\"(max-width: 1596px) 100vw, 1596px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1600\" height=\"849\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-37.png\" alt=\"\" class=\"wp-image-10274\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-37.png 1600w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-37-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-37-1024x543.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-37-768x408.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-37-535x284.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-37-1536x815.png 1536w\" sizes=\"(max-width: 1600px) 100vw, 1600px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-paragraph\">In this case we don&#8217;t have magic square of order 22 centered in magic square of order 14 having four equal sums double digits magic rectangles.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles and <br>Magic Squares of Order 22: First Type<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-paragraph\">Below are examples of <strong>magic squares of order 22<\/strong> centered in magic squares of order 6, 10 and 14 respectively. These are constructed with four equal sums <strong>cornered magic rectangles <\/strong>of orders <strong>8&#215;14, 6&#215;16<\/strong> and <strong>4&#215;18<\/strong> respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1597\" height=\"844\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-14.png\" alt=\"\" class=\"wp-image-10276\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-14.png 1597w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-14-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-14-1024x541.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-14-768x406.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-14-535x283.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-14-1536x812.png 1536w\" sizes=\"(max-width: 1597px) 100vw, 1597px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1594\" height=\"844\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-49.png\" alt=\"\" class=\"wp-image-10279\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-49.png 1594w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-49-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-49-1024x542.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-49-768x407.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-49-535x283.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-49-1536x813.png 1536w\" sizes=\"(max-width: 1594px) 100vw, 1594px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1596\" height=\"850\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-110-1.png\" alt=\"\" class=\"wp-image-10282\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-110-1.png 1596w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-110-1-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-110-1-1024x545.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-110-1-768x409.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-110-1-535x285.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-110-1-1536x818.png 1536w\" sizes=\"(max-width: 1596px) 100vw, 1596px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles and <br>Magic Squares of Order 22: Second Type<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-paragraph\">Below are examples of <strong>magic squares of order 22<\/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>8&#215;14 <\/strong>and <strong>6&#215;16<\/strong> respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1597\" height=\"847\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-16.png\" alt=\"\" class=\"wp-image-10277\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-16.png 1597w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-16-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-16-1024x543.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-16-768x407.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-16-535x284.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-16-1536x815.png 1536w\" sizes=\"(max-width: 1597px) 100vw, 1597px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1594\" height=\"845\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-66.png\" alt=\"\" class=\"wp-image-10280\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-66.png 1594w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-66-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-66-1024x543.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-66-768x407.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-66-535x284.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-66-1536x814.png 1536w\" sizes=\"(max-width: 1594px) 100vw, 1594px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">3. Cornered Magic Rectangles and <br>Magic Squares of Order 22: Third Type<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-paragraph\">Below are examples of <strong>magic squares of order 22<\/strong> centered in magic squares of order 6, 10 and 14 respectively. These are constructed with four equal sums <strong>cornered magic rectangles <\/strong>of orders <strong>8&#215;14, 6&#215;16<\/strong> and <strong>4&#215;18<\/strong> respectively.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1596\" height=\"849\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-22.png\" alt=\"\" class=\"wp-image-10278\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-22.png 1596w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-22-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-22-1024x545.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-22-768x409.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-22-535x285.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-22-1536x817.png 1536w\" sizes=\"(max-width: 1596px) 100vw, 1596px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1598\" height=\"846\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-77.png\" alt=\"\" class=\"wp-image-10281\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-77.png 1598w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-77-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-77-1024x542.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-77-768x407.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-77-535x283.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-77-1536x813.png 1536w\" sizes=\"(max-width: 1598px) 100vw, 1598px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1599\" height=\"846\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-129.png\" alt=\"\" class=\"wp-image-10283\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-129.png 1599w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-129-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-129-1024x542.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-129-768x406.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-129-535x283.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/MR-22x22-129-1536x813.png 1536w\" sizes=\"(max-width: 1599px) 100vw, 1599px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 53%,rgb(255,105,0) 100%)\">PDF File of Magic Squares of Order 22<\/h3>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">Below is a pdf file of 132 magic squares of order 22 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\/22x22.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of 22x22.\"><\/object><a id=\"wp-block-file--media-60b0ed1b-a33d-4154-ab2b-b7014de0b610\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/22x22.pdf\">22&#215;22<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/09\/22x22.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 class=\"wp-block-paragraph\"><\/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 class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ol class=\"has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background wp-block-list\">\n<li><strong>Inder J. Taneja<\/strong>, <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 class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Never Seen Before This work brings magic squares in very different way. These are based on three different types of magic rectangles:1. Bordered Magic Rectangles:2. Double Digits Magic Rectangles:3. Cornered magic rectangles. First of let&#8217;s understand these magic rectangles one by one. Removing external borders in each case still we left with lower order magic [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":10281,"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-10061","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-22x22-77.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10061","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=10061"}],"version-history":[{"count":8,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10061\/revisions"}],"predecessor-version":[{"id":10409,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10061\/revisions\/10409"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/10281"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10061"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10061"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10061"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}