{"id":12610,"date":"2024-09-29T23:28:38","date_gmt":"2024-09-30T02:28:38","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=12610"},"modified":"2024-11-06T00:31:50","modified_gmt":"2024-11-06T03:31:50","slug":"universal-and-upside-down-magic-squares-of-order-3-to-6","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=12610","title":{"rendered":"Universal and Upside-Down Magic Squares of Order 3 to 6"},"content":{"rendered":"\n<p>There are many ways of representing magic squares with palindromic type entries. Also, we can write magic squares in the composite forms based on pair of Latin squares. This paper works with magic squares of order 3 to 6. By <strong>upside-down<\/strong>, we understand than by making 180<sup>o<\/sup> it remains same. When the magic square is of both type, i.e., <strong>upside-down<\/strong> and <strong>mirror looking<\/strong>, we call it an <strong>universal<\/strong> magic square. By mirror looking we understand that putting in front of mirror, still we see the image as a magic square. In case of mirror looking, writing as <strong>digitais fonts<\/strong>, 2 becomes 5 and 5 as 2. In case of <strong>upside-down<\/strong>, 6 becomes 9 and 9 as 6. <\/p>\n\n\n\n<p>Follow the following link to see this work online.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Squares of Orders 3 to 6, <strong>Zenodo<\/strong>, November 05, 2024, pp. 1-61, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14041149\">https:\/\/doi.org\/10.5281\/zenodo.14041149<\/a>.<\/li>\n<\/ul>\n\n\n\n<p>For more work up to order 25 see the reference list given at the end of this work.<br><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(42,222,179) 53%,rgb(155,81,224) 100%)\">Magic Squares of Order 3<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-3eeaf5a250fc3e78b1e247748c6bf37c\">a) 2-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-5068317a8562322c101b8865727279c3\">Example 1. The Digits (1,6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"189\" height=\"204\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3x3-169.png\" alt=\"\" class=\"wp-image-12613\"\/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>upside-down<\/strong> <strong>semi-magic<\/strong> square of order 3 with 3-digits (1,6,9).  The magic sum is <strong>S<sub>3&#215;3<\/sub>(1,6,9):=176.<\/strong> This magic square becomes becomes <strong>semi-magic<\/strong> in case of <strong>mirror looking<\/strong>. <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-a8e6e4d4c86abcac2aef6e997b39925e\">Example 2. The Digits (2,5,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"189\" height=\"199\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3x3-258.png\" alt=\"\" class=\"wp-image-12614\"\/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>universal<\/strong> magic square of order 3 with 3-digits (2,5,8).  The magic sum is <strong>S<sub>3&#215;3<\/sub>(2,5,8):=165.<\/strong> This magic square becomes becomes <strong>semi-magic<\/strong> in case of <strong>mirror looking<\/strong>. <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-4d5cacbfb96ac9feef1043c49b5b648d\">b) 3-Digits Cell Entries<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-8393c536809b4059e4f2a645c1a3b78c\">Example 3. The Digits (1,6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"446\" height=\"282\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-169.png\" alt=\"\" class=\"wp-image-12612\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-169.png 446w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-169-300x190.png 300w\" sizes=\"(max-width: 446px) 100vw, 446px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>upside-down<\/strong> <strong>semi-magic<\/strong> square of order 3 with 3-digits (1,6,9).  In this case, the entries are <strong>palindromes.<\/strong> The magic sum is <strong>S<sub>3&#215;3<\/sub>(1,6,9):=1776.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-07aa0960f22767275a848221d0659a96\">Example 4. The Digits (2,5,8)<\/h3>\n\n\n\n<ul class=\"wp-block-list\"><\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"455\" height=\"255\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-258.png\" alt=\"\" class=\"wp-image-12611\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-258.png 455w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-258-300x168.png 300w\" sizes=\"(max-width: 455px) 100vw, 455px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>universal<\/strong> magic square of order 3 with 3-digits (2,5,8).  The magic sum is <strong>S<sub>3&#215;3<\/sub>(2,5,8):=1665.<\/strong> In this case, the entries are <strong>palindromes.<\/strong> This magic square becomes becomes <strong>semi-magic<\/strong> in case of <strong>mirror looking<\/strong>. <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-19ce98b0ac653648f3f36d60d6df67d7\">c) 4-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-2c510c0554495e205d7fd4f1fc794b27\">Example 5. The Digits (1,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"318\" height=\"177\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-3x3-1.png\" alt=\"\" class=\"wp-image-13395\" style=\"width:464px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-3x3-1.png 318w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-3x3-1-300x167.png 300w\" sizes=\"(max-width: 318px) 100vw, 318px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>universal semi-magic<\/strong> square with <strong>semi-magic<\/strong> sum: S<sub>3&#215;3<\/sub>(1,8):=11110.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-e494f5f8a56d4fbb1292fcb21627b736\">Example 6. The Digits (2,5)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"339\" height=\"197\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-3x3-1.png\" alt=\"\" class=\"wp-image-13396\" style=\"width:464px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-3x3-1.png 339w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-3x3-1-300x174.png 300w\" sizes=\"(max-width: 339px) 100vw, 339px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>universal semi-magic<\/strong> square with <strong>semi-magic<\/strong> sum: S<sub>3&#215;3<\/sub>(2,5):=9999. This sum refer to original and upside-down versions. For mirror looking version the <strong>semi-magic<\/strong> sum is  S<sub>3&#215;3<\/sub>(2,5):=13332.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-3234df9d180effda6f802fbb5091ca43\">Example 7. The Digits (6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"371\" height=\"194\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-3x3-1.png\" alt=\"\" class=\"wp-image-13397\" style=\"width:464px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-3x3-1.png 371w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-3x3-1-300x157.png 300w\" sizes=\"(max-width: 371px) 100vw, 371px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>upside-down semi-magic<\/strong> square with <strong>semi-magic<\/strong> sum: S<sub>3&#215;3<\/sub>(6,9):=23634.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(42,222,179) 53%,rgb(155,81,224) 100%)\">Magic Squares of Order 4<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-3eeaf5a250fc3e78b1e247748c6bf37c\">a) 2-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-fc48b7b3eddf81e5791fcb9f44f77318\">Example 8. The Digits (1,6,8,9)<\/h3>\n\n\n\n<ul class=\"wp-block-list\"><\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"219\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/4x4-1689.png\" alt=\"\" class=\"wp-image-12615\"\/><\/figure>\n<\/div>\n\n\n<p>It is <strong>upside-down<\/strong> magic square of order 4 with 4-digits (1,6,8,9). The magic sum is <strong>S<sub>4&#215;4<\/sub>(1,6,8,9):=264.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-3ddacf565f39e8cfb75d3b9ce1f29dbc\">Example 9. The Digits (0,1,2,5)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"299\" height=\"198\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/4x4-0125.png\" alt=\"\" class=\"wp-image-12616\" style=\"width:299px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>It is <strong>universal <\/strong> magic square of order 4 with 4-digits (0,1,2,5). The magic sum is <strong>S<sub>4&#215;4<\/sub>(0,1,2,5):=88.<\/strong> <\/p>\n\n\n\n<p>It is well-known author&#8217;s stamp published at Macau &#8211; China &#8211; 2015. See below:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><img loading=\"lazy\" decoding=\"async\" width=\"225\" height=\"300\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/Selo-1-225x300.jpg\" alt=\"\" class=\"wp-image-12621\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/Selo-1-225x300.jpg 225w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/Selo-1-769x1024.jpg 769w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/Selo-1-768x1022.jpg 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/Selo-1-535x712.jpg 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/Selo-1-1154x1536.jpg 1154w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/Selo-1.jpg 1480w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-2d8472a6410210099c07b9f98950feb9\">b) 3-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-78ba8abc119bef3c9ca42cd8ad3c0145\">Example 10. The Digits (1,6,8,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"241\" height=\"148\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/4-1689.png\" alt=\"\" class=\"wp-image-12618\" style=\"width:332px;height:auto\"\/><\/figure>\n<\/div>\n\n\n<p>It is <strong>upside-down<\/strong> magic square of order 4 with 4-digits (1,6,8,9). In this case, the entries are <strong>palindromes<\/strong>. The magic sum is <strong>S<sub>4&#215;4<\/sub>(1,6,8,9):=2664.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-485bdad8edbb1a15c5888710a6bc6e53\">Example 11. The Digits (1,2,5,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"344\" height=\"221\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/4x4-1258.png\" alt=\"\" class=\"wp-image-12619\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/4x4-1258.png 344w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/4x4-1258-300x193.png 300w\" sizes=\"(max-width: 344px) 100vw, 344px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>universal <\/strong>magic square of order 4 with 4-digits (1,2,5,8). In this case, the entries are <strong>palindromes<\/strong>. The magic sum is <strong>S<sub>4&#215;4<\/sub>(1,2,5,8):=1776.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-19ce98b0ac653648f3f36d60d6df67d7\">c) 4-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-400157a3a66b512900f78b1bdb010ec9\">Example 12. The Digits (1,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"583\" height=\"257\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-18a.png\" alt=\"\" class=\"wp-image-12632\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-18a.png 583w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-18a-300x132.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-18a-535x236.png 535w\" sizes=\"(max-width: 583px) 100vw, 583px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>pandiagonal universal <\/strong>magic square of order 4 with 2-digits (1,8). The magic sum is <strong>S<sub>4&#215;4<\/sub>(1,8):=19998.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-d9848fcd2a70e570c4aa8969e896e5a6\">Example 13. The Digits (2,5)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"685\" height=\"307\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-25.png\" alt=\"\" class=\"wp-image-12624\" style=\"width:596px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-25.png 685w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-25-300x134.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-25-535x240.png 535w\" sizes=\"(max-width: 685px) 100vw, 685px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>universal <\/strong>magic square of order 4 with 2-digits (2,5). The magic sum is <strong>S<sub>4&#215;4<\/sub>(2,5):=15554.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-3b91b29acfbcf07ab2fd43daf08cfc19\">Example 14. The Digits (6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"653\" height=\"343\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-69.png\" alt=\"\" class=\"wp-image-12622\" style=\"width:553px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-69.png 653w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-69-300x158.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/2-69-535x281.png 535w\" sizes=\"(max-width: 653px) 100vw, 653px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>upside-down<\/strong> magic square of order 4 with 2-digits (6,9). The magic sum is <strong>S<sub>4&#215;4<\/sub>(6,9):=33330.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-c596dd9933a6d2b8542c895d0a304718\">d) 7-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-2dab42b017cbdccd4b724768810baf4f\">Example 15. The Digits (1,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1028\" height=\"318\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-18.png\" alt=\"\" class=\"wp-image-12629\" style=\"width:789px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-18.png 1028w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-18-300x93.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-18-1024x317.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-18-768x238.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-18-535x165.png 535w\" sizes=\"(max-width: 1028px) 100vw, 1028px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>pandiagonal universal <\/strong>magic square of order 4 with 2-digits (1,8). The entries are <strong>palindromes<\/strong>. The magic sum is <strong>S<sub>4&#215;4<\/sub>(1,8):=19999998.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-1ed4016c098ec5ca93a6f15417825e3d\">Example 16. The Digits (2, 5)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"980\" height=\"305\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-25.png\" alt=\"\" class=\"wp-image-12630\" style=\"width:794px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-25.png 980w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-25-300x93.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-25-768x239.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-25-535x167.png 535w\" sizes=\"(max-width: 980px) 100vw, 980px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal universal <\/strong>magic square of order 4 with 2-digits (2,5). The entries are <strong>palindromes<\/strong>. The magic sum is <strong>S<sub>4&#215;4<\/sub>(2,5):=15555554.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-332c4b720aecece2029de5a67ffa4729\">Example 17. The Digits (6, 9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1032\" height=\"342\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-69.png\" alt=\"\" class=\"wp-image-12628\" style=\"width:766px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-69.png 1032w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-69-300x99.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-69-1024x339.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-69-768x255.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/7-69-535x177.png 535w\" sizes=\"(max-width: 1032px) 100vw, 1032px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>upside-down<\/strong> magic square of order 4 with 2-digits (6,9). The entries are <strong>palindromes<\/strong>. The magic sum is <strong>S<sub>4&#215;4<\/sub>(6,9):=33333330.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(42,222,179) 53%,rgb(155,81,224) 100%)\">Magic Squares of Order 5<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-621b74c0715e70e5ac4876266596be56\">a) 5-Digits Cell Entries<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-9427bbb6f762521d60d6e6e2efb8f299\">Example 18. The Digits (0,1,6,8,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"418\" height=\"418\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-01689.png\" alt=\"\" class=\"wp-image-12640\" style=\"width:352px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-01689.png 418w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-01689-300x300.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-01689-150x150.png 150w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-01689-256x256.png 256w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>upside-down<\/strong> magic square of order 5 with 5-digits (0,1,6,8,9). The magic sum is <strong>S<sub>5&#215;5<\/sub>(0,1,6,8,9):=264.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-4861605059abefb2c3d35b69d2e4b29c\">Example 19. The Digits (2,5,6,8,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"485\" height=\"351\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/6-25689.png\" alt=\"\" class=\"wp-image-12634\" style=\"width:358px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/6-25689.png 485w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/6-25689-300x217.png 300w\" sizes=\"(max-width: 485px) 100vw, 485px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>upside-down<\/strong> magic square of order 5 with 5-digits (2,5,6,8,9). The magic sum is <strong>S<sub>5&#215;5<\/sub>(2,5,6,8,9):=330.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-3604bf8ab7294d835b25e036bb7388ce\">Example 20. The Digits (0,1,2,5,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"375\" height=\"279\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-2d-01258.png\" alt=\"\" class=\"wp-image-12633\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-2d-01258.png 375w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-2d-01258-300x223.png 300w\" sizes=\"(max-width: 375px) 100vw, 375px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>universal<\/strong> magic square of order 5 with 5-digits (0,1,2,5,8). The magic sum is <strong>S<sub>5&#215;5<\/sub>(0,1,2,5,8):=176.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-4d5cacbfb96ac9feef1043c49b5b648d\">b) 3-Digits Cell Entries<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-ea310ae325daaf6da6b4e775302ad872\">Example 21. The Digit (1,2,5,6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"527\" height=\"270\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5x5-12569.png\" alt=\"\" class=\"wp-image-12644\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5x5-12569.png 527w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5x5-12569-300x154.png 300w\" sizes=\"(max-width: 527px) 100vw, 527px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>upside-down<\/strong> magic square of order 5 with 5-digits (1,2,5,6,9). In this case the entries are <strong>palindromes<\/strong>. The magic sum is <strong>S<sub>5&#215;5<\/sub>(1,2,5,6,9):=2553.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-c4749083645c254301e02c0166b57e93\">Example 22. The Digits (0,1,2,5,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"534\" height=\"280\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5x5-01258.png\" alt=\"\" class=\"wp-image-12642\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5x5-01258.png 534w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5x5-01258-300x157.png 300w\" sizes=\"(max-width: 534px) 100vw, 534px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>universal <\/strong>magic square of order 5 with 5-digits (0,1,2,5,8). In this case the entries are <strong>palindromes<\/strong>. The magic sum is <strong>S<sub>5&#215;5<\/sub>(0,1,2,5,8):=1776.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-46d4ab2df474ba39ad3e65d5385e51ba\">c) 4-Digits Cell Entries<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-ecad6631b4a7e2fcfc5ca5fbdfc2ed28\">Example 23. The Digits (1,6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"814\" height=\"409\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-169a.png\" alt=\"\" class=\"wp-image-12645\" style=\"width:778px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-169a.png 814w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-169a-300x151.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-169a-768x386.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-169a-535x269.png 535w\" sizes=\"(max-width: 814px) 100vw, 814px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>upside-down<\/strong> magic square of order 5 with 3-digits (1,6,9). The magic sum is <strong>S<sub>5&#215;5<\/sub>(1,6,9):=34441.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-2d86406a11f50ac28ff1ce285f54be23\">Example 24. The Digits (6,8,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"553\" height=\"340\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-689a.png\" alt=\"\" class=\"wp-image-12646\" style=\"width:785px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-689a.png 553w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-689a-300x184.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/5-689a-535x329.png 535w\" sizes=\"(max-width: 553px) 100vw, 553px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>upside-down<\/strong> magic square of order 5 with 3-digits (6,8,9). The magic sum is <strong>S<sub>5&#215;5<\/sub>(6,8,9):=42218.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-6140b45afa95ec155ade211eac0847af\">Example 25. The Digits (2,5,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"781\" height=\"413\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-258a.png\" alt=\"\" class=\"wp-image-12647\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-258a.png 781w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-258a-300x159.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-258a-768x406.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/3-258a-535x283.png 535w\" sizes=\"(max-width: 781px) 100vw, 781px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>universal <\/strong> magic square of order 5 with 3-digits (2,5,8). The magic sum is <strong>S<sub>5&#215;5<\/sub>(2,5,8):=24442.<\/strong> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-ec61b07d51a39014f30f38c3524f9bd3\">b) 6-Digits Cell Entries<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-348190a8ece719ea2cc8822c92702064\">Example 26. The Digits (1,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"690\" height=\"292\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-5x5-2.png\" alt=\"\" class=\"wp-image-13398\" style=\"width:906px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-5x5-2.png 690w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-5x5-2-300x127.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-5x5-2-535x226.png 535w\" sizes=\"(max-width: 690px) 100vw, 690px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>universal <\/strong> magic square of order 5 with 2-digits (1,8). The magic sum is <strong>S<sub>5&#215;5<\/sub>(1,8):=2958956<\/strong>.  <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-67e07373c3cfbf9164c3a3145d1d1f19\">Example 27. The Digits (2,5)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"722\" height=\"310\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-5x5-1.png\" alt=\"\" class=\"wp-image-13399\" style=\"width:906px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-5x5-1.png 722w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-5x5-1-300x129.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-5x5-1-535x230.png 535w\" sizes=\"(max-width: 722px) 100vw, 722px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>universal <\/strong> magic square of order 5 with 2-digits (1,8). The magic sum of original and upside versions are same, i.e., <strong>S<sub>5&#215;5<\/sub>(2,5):=1747746<\/strong>. The magic sum of mirror looking version is <strong>S<sub>5&#215;5<\/sub>(2,5):=2141139. <\/strong>It is <strong>pandiagonal<\/strong> in all the three cases.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-aeaf487421fd0d47bb197909ad4b1c33\">Example 27. The Digits (6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"805\" height=\"314\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-5x5-1.png\" alt=\"\" class=\"wp-image-13400\" style=\"width:906px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-5x5-1.png 805w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-5x5-1-300x117.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-5x5-1-768x300.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-5x5-1-535x209.png 535w\" sizes=\"(max-width: 805px) 100vw, 805px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>pandiagonal<\/strong> <strong>universal <\/strong> magic square of order 5 with 2-digits (6,9). The magic sum is <strong>S<sub>5&#215;5<\/sub>(6,9):=2958956<\/strong>. The magic sum of <strong>upside-down<\/strong> version is different and is given by <strong>S<sub>5&#215;5<\/sub>(6,9):=4363359.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(42,222,179) 53%,rgb(155,81,224) 100%)\">Magic Squares of Order 6<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-b3fd78649fe4d2408410d66cfcf938a5\">a) 4-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-d2e86a9544bc8ebd2376b68f5ca6ca47\">Example 28. The Digits (2,5,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"951\" height=\"441\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/6-258a.png\" alt=\"\" class=\"wp-image-12648\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/6-258a.png 951w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/6-258a-300x139.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/6-258a-768x356.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/6-258a-535x248.png 535w\" sizes=\"(max-width: 951px) 100vw, 951px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>upside-down<\/strong> magic square of order 5 with 3-digits (6,8,9). The magic sum is <strong>S<sub>6&#215;6<\/sub>(2,5,8):=33330.<\/strong> This magic square is only <strong>upside-down<\/strong> because its<strong> mirror looking image<\/strong> is no more a<strong> <\/strong>magic square.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-8d524e11f2c4b1bc44f3d10af8095dd1\">a) 6-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-971d96130c0e44bb9b6c510a03551936\">Example 29. The Digits (1,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"826\" height=\"345\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-6x6-1.png\" alt=\"\" class=\"wp-image-13401\" style=\"width:902px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-6x6-1.png 826w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-6x6-1-300x125.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-6x6-1-768x321.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/18-6x6-1-535x223.png 535w\" sizes=\"(max-width: 826px) 100vw, 826px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>universal <\/strong>magic square of order 6 in 2-digits 1 and 8. The magic sums is <strong>S<sub>6&#215;6<\/sub>(1,8):=2999997<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-69332aea5bf16377e5f7db7eccae131c\">Example 30. The Digits (2,5)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"862\" height=\"374\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-6x6-1.png\" alt=\"\" class=\"wp-image-13402\" style=\"width:902px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-6x6-1.png 862w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-6x6-1-300x130.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-6x6-1-768x333.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/25-6x6-1-535x232.png 535w\" sizes=\"(max-width: 862px) 100vw, 862px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>universal <\/strong>magic square of order 6 in 2-digits 2 and 5. The magic sum in all the three cases is same, i.e., <strong>S<sub>6&#215;6<\/sub>(2,5):=2333331<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-c01af265381c3f4c2e106010932d591e\">Example 31. The Digits (6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"895\" height=\"373\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-6x6-1.png\" alt=\"\" class=\"wp-image-13403\" style=\"width:902px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-6x6-1.png 895w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-6x6-1-300x125.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-6x6-1-768x320.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/09\/69-6x6-1-535x223.png 535w\" sizes=\"(max-width: 895px) 100vw, 895px\" \/><\/figure>\n<\/div>\n\n\n<p>It is <strong>universal <\/strong>magic square of order 6 in 2-digits 6 and 9. The magic sums is <strong>S<sub>6&#215;6<\/sub>(6,9):=4999995<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(42,222,179) 53%,rgb(155,81,224) 100%)\">References<\/h3>\n\n\n\n<ol class=\"wp-block-list has-light-green-cyan-to-vivid-green-cyan-gradient-background has-background\">\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Squares of Orders 3 to 6, <strong>Zenodo<\/strong>, November 05, 2024, pp. 1-61, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14041149\">https:\/\/doi.org\/10.5281\/zenodo.14041149<\/a>\n<ul class=\"wp-block-list\">\n<li>Site link: <a href=\"https:\/\/numbers-magic.com\/?p=12610\">Universal and Upside-Down Magic Squares of Order 3 to 6 (new site)<\/a><\/li>\n\n\n\n<li>Site link: <a href=\"https:\/\/inderjtaneja.wordpress.com\/2024\/09\/30\/universal-and-upside-down-magic-squares-of-order-3-to-6\/\">Universal and Upside-Down Magic Squares of Order 3 to 6 (old site)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Squares of Orders 7 to 10, <strong>Zenodo<\/strong>, November 05, 2024, pp. 1-120, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14041164\">https:\/\/doi.org\/10.5281\/zenodo.14041164<\/a>\n<ul class=\"wp-block-list\">\n<li>Site link: <a href=\"https:\/\/numbers-magic.com\/?p=12654\">Universal and Upside-Down Magic Squares of Orders 7 to 10 (new site)<\/a><\/li>\n\n\n\n<li>Site link: <a href=\"https:\/\/inderjtaneja.wordpress.com\/2024\/09\/30\/universal-and-upside-down-magic-squares-of-orders-7-to-10\/\">Universal and Upside-Down Magic Squares of Orders 7 to 10 (old site)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Squares of Orders 11 to 15, <strong>Zenodo<\/strong>, October 06, 2024, pp. 1-141, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14041168\">https:\/\/doi.org\/10.5281\/zenodo.14041168<\/a>\n<ul class=\"wp-block-list\">\n<li>Site link: <a href=\"https:\/\/numbers-magic.com\/?p=12731\">Universal and Upside-Down Magic Squares of Orders 11 to 15 (new site)<\/a><\/li>\n\n\n\n<li>Site link: <a href=\"https:\/\/inderjtaneja.wordpress.com\/2024\/10\/07\/universal-and-upside-down-magic-squares-of-orders-11-to-15\/\">Universal and Upside-Down Magic Squares of Orders 11 to 15 (old site)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Squares of Order 16, <strong>Zenodo<\/strong>, October 16, 2024, pp. 1-28, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.13942620\">https:\/\/doi.org\/10.5281\/zenodo.13942620<\/a>\n<ul class=\"wp-block-list\">\n<li>Site link: <a href=\"https:\/\/numbers-magic.com\/?p=13138\">Universal and Upside-Down Magic and Bimagic Squares of Order 16 (new site) <\/a><\/li>\n\n\n\n<li>Site link: <a href=\"https:\/\/inderjtaneja.wordpress.com\/2024\/10\/16\/universal-and-upside-down-magic-and-bimagic-squares-of-order-16\/\">Universal and Upside-Down Magic and Bimagic Squares of Order 16 (old site)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Squares of Order 20, <strong>Zenodo<\/strong>, October 20, 2024, pp. 1-56, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.13958700\">https:\/\/doi.org\/10.5281\/zenodo.13958700<\/a>.\n<ul class=\"wp-block-list\">\n<li>Site link: <a href=\"https:\/\/numbers-magic.com\/?p=13186\">Universal and Upside-Down Magic Squares of Order 20 (new site)<\/a><\/li>\n\n\n\n<li>Site link: <a href=\"https:\/\/inderjtaneja.wordpress.com\/2024\/10\/23\/universal-and-upside-down-magic-and-bimagic-squares-of-order-16-2\/\">Universal and Upside-Down Magic Squares of Order 20 (old site)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Squares of Order 21, <strong>Zenodo<\/strong>, October 23, 2024, pp. 1-49, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.13982859\">https:\/\/doi.org\/10.5281\/zenodo.13982859<\/a>\n<ul class=\"wp-block-list\">\n<li>Site link: <a href=\"https:\/\/numbers-magic.com\/?p=13242\">Universal and Upside-Down Magic Squares of Order 21 (new site)<\/a><\/li>\n\n\n\n<li>Site link: <a href=\"https:\/\/inderjtaneja.wordpress.com\/2024\/10\/23\/10984\/\">Universal and Upside-Down Magic Squares of Order 21 (old site)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Squares of Order 24, <strong>Zenodo<\/strong>, October 29, 2024, pp. 1-82, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14004788\">https:\/\/doi.org\/10.5281\/zenodo.14004788<\/a>\n<ul class=\"wp-block-list\">\n<li>Site link: <a href=\"https:\/\/numbers-magic.com\/?p=13299\">Universal and Upside-Down Magic Squares of Order 24 (new site)<\/a>.<\/li>\n\n\n\n<li>Site link: <a href=\"https:\/\/inderjtaneja.wordpress.com\/2024\/10\/29\/universal-and-upside-down-magic-squares-of-order-24\/\">Universal and Upside-Down Magic Squares of Order 24 (old site)<\/a><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic and Bimagic Squares of Order 25, <strong>Zenodo<\/strong>, October 30, 2024, pp. 1-53, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14014851\">https:\/\/doi.org\/10.5281\/zenodo.14014851<\/a>.\n<ul class=\"wp-block-list\">\n<li>Site link: <a href=\"https:\/\/numbers-magic.com\/?p=13336\">Universal and Upside-Down Magic and Bimagic Squares of Order 25 (new site) <\/a><\/li>\n\n\n\n<li>Site link: <a href=\"https:\/\/inderjtaneja.wordpress.com\/2024\/10\/30\/universal-and-upside-down-magic-and-bimagic-squares-of-order-25\/\">Universal and Upside-Down Magic and Bimagic Squares of Order 25 (old site)<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>There are many ways of representing magic squares with palindromic type entries. Also, we can write magic squares in the composite forms based on pair of Latin squares. This paper works with magic squares of order 3 to 6. By upside-down, we understand than by making 180o it remains same. When the magic square is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":12630,"comment_status":"open","ping_status":"closed","sticky":false,"template":"elementor_theme","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3],"tags":[],"class_list":["post-12610","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\/2024\/09\/7-25.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/12610","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=12610"}],"version-history":[{"count":19,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/12610\/revisions"}],"predecessor-version":[{"id":13409,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/12610\/revisions\/13409"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/12630"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12610"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12610"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12610"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}