{"id":9988,"date":"2023-08-06T13:38:57","date_gmt":"2023-08-06T16:38:57","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=9988"},"modified":"2023-08-09T02:26:15","modified_gmt":"2023-08-09T05:26:15","slug":"double-digits-bordered-magic-squares-of-orders-7-to-108","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=9988","title":{"rendered":"New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><strong><em><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">Never seen before.<\/mark><\/em><\/strong><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-justify has-blush-light-purple-gradient-background has-background\">By boarded magic squares we understand that if we remove the external border still we are left with magic squares of lower orders with sequential entries. Single digits bordered magic squares are well-known. This work is for double digits not non in the literature. We are bringing this concept for the first time in the literature.  During these days some studies in this directions are made. These are given below in seperate parts. One for <strong>even number orders<\/strong> and another for <strong>odd number orders<\/strong>. Most of the work is done manually. A special trick is applied to bring all these magic squares.<br><br>Lower order double digits magic squares can also be calculated using software by Harry While: <a href=\"https:\/\/budshaw.ca\/Download.html\">Downloads (budshaw.ca)<\/a>. <\/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) 36%,rgb(255,105,0) 100%)\">Double Digits Bordered Magic Squares: <br><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">Even Number Orders<\/mark><\/h3>\n\n\n\n<p><\/p>\n\n\n\n<ul class=\"has-blush-light-purple-gradient-background has-background wp-block-list\">\n<li>The work for order of type 4k, k=2, 3, &#8230; i.e., for orders 8, 12, 16, 20, 24, 28, 32, 36 and 40:<br>\n<ul class=\"wp-block-list\">\n<li>Inder J. Taneja, Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24, <strong>Zenodo<\/strong>, April 26, 2023, pp. 1-43, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.7866956\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.7866956<\/a>.<br>Site Link: <a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=8493\" target=\"_blank\">Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24 .<\/a><br><\/li>\n\n\n\n<li>Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 28 and 32, <strong>Zenodo<\/strong>, April 26, 2023, pp. 1-36, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.7866981\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.7866981<\/a>.<br>Site Link: <a href=\"https:\/\/numbers-magic.com\/?p=8551\">Two Digits Bordered Magic Squares of Orders 28 and 32<\/a>.<br><\/li>\n\n\n\n<li>Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 36 and 40, <strong>Zenodo<\/strong>, May 04, 2023, pp. 1-41, <a rel=\"noreferrer noopener\" href=\"https:\/\/doi.org\/10.5281\/zenodo.7896709\" target=\"_blank\">https:\/\/doi.org\/10.5281\/zenodo.7896709<\/a>.<br>Site Links: <br>1. <a href=\"https:\/\/numbers-magic.com\/?p=8669\">Two digits Bordered Magic Squares of Order 36<\/a>.<br>2. <a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=8687\" target=\"_blank\">Two digits Bordered Magic Squares of Order 40<\/a><a href=\"https:\/\/numbers-magic.com\/?p=8687\" target=\"_blank\" rel=\"noreferrer noopener\">.<\/a><a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=8687\" target=\"_blank\"> <\/a><br><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>The work is for the order of type 4k+2, k=2, 3, &#8230;, i.e., for the orders 10, 14, 18, 22, 26, 30, 34 and 38:<br>\n<ul class=\"wp-block-list\">\n<li>Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 10, 14, 18 and 22, <strong>Zenodo<\/strong>, April 30, 2023, pp. 1-43, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.7880931\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.7880931<\/a>. <br>Site Link: <a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=8493\" target=\"_blank\"><a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=8583\" target=\"_blank\">Two digits Bordered Magic Squares of Orders 10, 14, 18 and 22<\/a>.<\/a><br><\/li>\n\n\n\n<li>Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 26 and 30, <strong>Zenodo<\/strong>, April 30, 2023, pp. 1-45, https:\/\/doi.org\/10.5281\/zenodo.7880937.<br>Site Link: <a href=\"https:\/\/numbers-magic.com\/?p=8551\"><a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=8583\" target=\"_blank\"><a href=\"https:\/\/numbers-magic.com\/?p=8629\">Two digits Bordered Magic Squares of Orders 26 and 30<\/a>.<\/a><\/a><br><\/li>\n\n\n\n<li>Inder J. Taneja, Two digits Bordered Magic Squares of Orders 34 and 38, <strong>Zenodo<\/strong>, May 10, 2023, pp. 1-45, <a rel=\"noreferrer noopener\" href=\"https:\/\/doi.org\/10.5281\/zenodo.7922571\" target=\"_blank\">https:\/\/doi.org\/10.5281\/zenodo.7922571<\/a>.<br>Site Links:<br>1. <a href=\"https:\/\/numbers-magic.com\/?p=8669\"><\/a><a href=\"https:\/\/numbers-magic.com\/?p=8838\">Two digits Bordered Magic Squares of Order 34<\/a>.<br>2. <a href=\"https:\/\/numbers-magic.com\/?p=8856\">Two digits Bordered Magic Squares of Order 38<\/a><a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=8856\" target=\"_blank\">.<\/a><\/li>\n<\/ul>\n<\/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) 36%,rgb(255,105,0) 100%)\">Double Digits Bordered Magic Squares:<br><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">Odd Number Orders<\/mark><\/h3>\n\n\n\n<ul class=\"has-blush-light-purple-gradient-background has-background wp-block-list\">\n<li>The work for double digits odd order bordered magic squares:<br>For the previous work on&nbsp;<strong>odd orders from 3 to 31<\/strong>&nbsp;see the following links:<br>\n\n\n\n\n\n\n\n\n\n\n\n\n<ul class=\"wp-block-list\">\n<li>Inder J. Taneja, Odd Order Magic Squares: Orders 3 to 15, Zenodo, June 15, 2023, pp. 1-43, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8043030\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.8043030<\/a>.<br>Site Link: <a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=9215\" target=\"_blank\">Odd Order Magic Squares: Orders 3 to 15<\/a><br><\/li>\n\n\n\n<li>Inder J. Taneja, Magic Squares of Orders 17 and 19, Zenodo, June 15, 2023, pp. 1-38, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8043105\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.8043105<\/a>.<br>Site Link: <a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=9274\" target=\"_blank\">Magic Squares of Orders 17 and 19<\/a><br><\/li>\n\n\n\n<li>Inder J. Taneja, Magic Squares of Orders 21 and 23, Zenodo, June 15, 2023, pp. 1-43, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8043198\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.8043198<\/a>.<br>Site Links: <a href=\"https:\/\/numbers-magic.com\/?p=9326\">Magic Squares of Order 21<\/a>. <br><\/li>\n\n\n\n<li>Inder J. Taneja, Magic Squares of Orders 21 and 23, Zenodo, June 15, 2023, pp. 1-43, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8043198\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.8043198<\/a>.<br>Site Links: <a href=\"https:\/\/numbers-magic.com\/?p=9349\">Magic Squares of Order 23<\/a>.<br><\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Inder J. Taneja, Magic Squares of Order 25, Zenodo, June 15, 2023, pp. 1-27, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8043228\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.8043228<\/a>.<br>Site Links: <a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=9373\" target=\"_blank\">Magic Squares of Order 25<\/a>.<br><\/li>\n\n\n\n<li>Inder J. Taneja, Magic Squares of Order 27, Zenodo, August 06, 2023, pp. 1-32, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8218291\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.8218291<\/a>.<br>Site Links: <a rel=\"noreferrer noopener\" href=\"https:\/\/numbers-magic.com\/?p=9444\" target=\"_blank\">Magic Squares of Order 27.<\/a><br><\/li>\n\n\n\n<li>Inder J. Taneja, Magic Squares of Order 29, Zenodo, August 06, 2023, pp. 1-30, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8218771\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.8218771<\/a>.<br>Site Links: <a href=\"https:\/\/numbers-magic.com\/?p=9468\">Magic Squares of Order 29.<\/a> <br><\/li>\n\n\n\n<li>Inder J. Taneja, Magic Squares of Order 31, Zenodo, August 06, 2023, pp. 1-35, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.8219053\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/doi.org\/10.5281\/zenodo.8219053<\/a>.<br>Site Links: <a href=\"https:\/\/numbers-magic.com\/?p=9493\">Magic Squares of Order 31<\/a>.<br><br><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-blush-light-purple-gradient-background has-background\">The above work is done as extended study for each order. This work of double digits magic squares just given one example in each case for orders 7 to 108. See below few examples. <br><br>In the end<strong> excel files <\/strong>of completed work are also attached for <strong>download<\/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(74,234,220) 0%,rgb(158,209,119) 20%,rgb(207,42,186) 40%,rgb(147,237,43) 60%,rgb(251,105,98) 80%,rgb(254,248,76) 100%)\">Double Digits Bordered Magic Squares: Orders 7 to 30<\/h3>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"1380\" height=\"375\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-7-8-9.png\" alt=\"\" class=\"wp-image-9989\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-7-8-9.png 1380w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-7-8-9-300x82.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-7-8-9-1024x278.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-7-8-9-768x209.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-7-8-9-535x145.png 535w\" sizes=\"(max-width: 1380px) 100vw, 1380px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"1854\" height=\"473\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-10-11-12.png\" alt=\"\" class=\"wp-image-9990\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-10-11-12.png 1854w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-10-11-12-300x77.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-10-11-12-1024x261.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-10-11-12-768x196.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-10-11-12-535x136.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-10-11-12-1536x392.png 1536w\" sizes=\"(max-width: 1854px) 100vw, 1854px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"1821\" height=\"549\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-13-14.png\" alt=\"\" class=\"wp-image-9991\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-13-14.png 1821w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-13-14-300x90.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-13-14-1024x309.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-13-14-768x232.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-13-14-535x161.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-13-14-1536x463.png 1536w\" sizes=\"(max-width: 1821px) 100vw, 1821px\" \/><\/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=\"2039\" height=\"610\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-15-16.png\" alt=\"\" class=\"wp-image-9992\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-15-16.png 2039w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-15-16-300x90.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-15-16-1024x306.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-15-16-768x230.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-15-16-535x160.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-15-16-1536x460.png 1536w\" sizes=\"(max-width: 2039px) 100vw, 2039px\" \/><\/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=\"2262\" height=\"683\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-17-18.png\" alt=\"\" class=\"wp-image-9993\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-17-18.png 2262w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-17-18-300x91.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-17-18-1024x309.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-17-18-768x232.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-17-18-535x162.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-17-18-1536x464.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-17-18-2048x618.png 2048w\" sizes=\"(max-width: 2262px) 100vw, 2262px\" \/><\/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=\"2416\" height=\"749\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-19-20.png\" alt=\"\" class=\"wp-image-9994\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-19-20.png 2416w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-19-20-300x93.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-19-20-1024x317.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-19-20-768x238.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-19-20-535x166.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-19-20-1536x476.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-19-20-2048x635.png 2048w\" sizes=\"(max-width: 2416px) 100vw, 2416px\" \/><\/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=\"1326\" height=\"784\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-21.png\" alt=\"\" class=\"wp-image-9995\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-21.png 1326w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-21-300x177.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-21-1024x605.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-21-768x454.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-21-535x316.png 535w\" sizes=\"(max-width: 1326px) 100vw, 1326px\" \/><\/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=\"1379\" height=\"819\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-22.png\" alt=\"\" class=\"wp-image-9996\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-22.png 1379w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-22-300x178.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-22-1024x608.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-22-768x456.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-22-535x318.png 535w\" sizes=\"(max-width: 1379px) 100vw, 1379px\" \/><\/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=\"1433\" height=\"851\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-23.png\" alt=\"\" class=\"wp-image-9997\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-23.png 1433w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-23-300x178.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-23-1024x608.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-23-768x456.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-23-535x318.png 535w\" sizes=\"(max-width: 1433px) 100vw, 1433px\" \/><\/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=\"1497\" height=\"887\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-24.png\" alt=\"\" class=\"wp-image-9998\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-24.png 1497w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-24-300x178.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-24-1024x607.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-24-768x455.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-24-535x317.png 535w\" sizes=\"(max-width: 1497px) 100vw, 1497px\" \/><\/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=\"1557\" height=\"924\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-25.png\" alt=\"\" class=\"wp-image-9999\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-25.png 1557w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-25-300x178.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-25-1024x608.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-25-768x456.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-25-535x317.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-25-1536x912.png 1536w\" sizes=\"(max-width: 1557px) 100vw, 1557px\" \/><\/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=\"1610\" height=\"957\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-26.png\" alt=\"\" class=\"wp-image-10000\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-26.png 1610w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-26-300x178.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-26-1024x609.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-26-768x457.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-26-535x318.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-26-1536x913.png 1536w\" sizes=\"(max-width: 1610px) 100vw, 1610px\" \/><\/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=\"1673\" height=\"990\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-27.png\" alt=\"\" class=\"wp-image-10001\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-27.png 1673w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-27-300x178.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-27-1024x606.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-27-768x454.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-27-535x317.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-27-1536x909.png 1536w\" sizes=\"(max-width: 1673px) 100vw, 1673px\" \/><\/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=\"1971\" height=\"1027\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-28.png\" alt=\"\" class=\"wp-image-10002\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-28.png 1971w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-28-300x156.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-28-1024x534.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-28-768x400.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-28-535x279.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-28-1536x800.png 1536w\" sizes=\"(max-width: 1971px) 100vw, 1971px\" \/><\/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=\"2147\" height=\"1061\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-29.png\" alt=\"\" class=\"wp-image-10003\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-29.png 2147w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-29-300x148.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-29-1024x506.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-29-768x380.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-29-535x264.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-29-1536x759.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-29-2048x1012.png 2048w\" sizes=\"(max-width: 2147px) 100vw, 2147px\" \/><\/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=\"2188\" height=\"1093\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-30.png\" alt=\"\" class=\"wp-image-10004\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-30.png 2188w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-30-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-30-1024x512.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-30-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-30-535x267.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-30-1536x767.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/dd-30-2048x1023.png 2048w\" sizes=\"(max-width: 2188px) 100vw, 2188px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>For more examples see the excel files below:<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-blush-light-purple-gradient-background has-background\">Excel files for Download<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>File 1: Odd Number Orders <\/strong><\/p>\n\n\n\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-23e43716-a946-4e5a-81bc-31cdff32ded5\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/Double-Digits-Odd-online.xlsx\">Double-Digits-Odd-online<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/Double-Digits-Odd-online.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-23e43716-a946-4e5a-81bc-31cdff32ded5\">Download<\/a><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>File 2: Even Number Orders <\/strong><\/p>\n\n\n\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-07ed6124-dffc-4999-b6c8-923dc4f7a552\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/Double-Digits-Even.xlsx\">Double-Digits-Even<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/08\/Double-Digits-Even.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-07ed6124-dffc-4999-b6c8-923dc4f7a552\">Download<\/a><\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Never seen before. By boarded magic squares we understand that if we remove the external border still we are left with magic squares of lower orders with sequential entries. Single digits bordered magic squares are well-known. This work is for double digits not non in the literature. We are bringing this concept for the first [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":9997,"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-9988","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\/08\/dd-23.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/9988","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=9988"}],"version-history":[{"count":4,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/9988\/revisions"}],"predecessor-version":[{"id":10015,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/9988\/revisions\/10015"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/9997"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9988"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9988"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9988"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}