{"id":13186,"date":"2024-10-20T13:55:59","date_gmt":"2024-10-20T16:55:59","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=13186"},"modified":"2024-11-05T14:59:43","modified_gmt":"2024-11-05T17:59:43","slug":"universal-and-upside-down-magic-squares-of-order-20","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=13186","title":{"rendered":"Universal and Upside-Down Magic Squares of Order 20"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">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 and bimagic squares of order 16. 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 as <strong>universal<\/strong> magic square. By <strong>mirror looking<\/strong>, we understand that putting in front of mirror, still we see the image as a magic square. In case of <strong>mirror looking<\/strong>, writing as <strong>digitais fonts<\/strong>, 2 becoms 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 class=\"wp-block-paragraph\">This work brings examples in different categories of entries. These are divided in three parts. In each part we worked with blocks of magic squares of orders 4 and 5.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>4-Digits Cell Entries<\/strong>\n<ul class=\"wp-block-list\">\n<li>In this case, the work is in 5-digits, such as, (0,1,6,8,9) and (0,1,2,5,8). 5-Digits combinations lead us to 4-digits cell entries.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>6-Digits Cell Entries<\/strong>\n<ul class=\"wp-block-list\">\n<li>In this case, the work is in 3-digits, such as, (1,6,9) and (2,5,8).  3-Digits combinations lead us to 6-digits cell entries.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>10-Digits Cell Entries<\/strong>\n<ul class=\"wp-block-list\">\n<li>In this case, the work is in 2-digits, such as, (1,8), (2,5) and (6,9). 2-Digits combinations lead us to 10-digits cell entries.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">For complete work on <strong>universal<\/strong> and <strong>upside-down magic, bimagic<\/strong> and<strong> pandiagonal<\/strong> squares  see the reference list at the end. For this work online refer the following link:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\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>. <\/li>\n<\/ul>\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%)\">Universal and Upside-Down Magic of Order 20<\/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-text-align-center\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">Upside-Down Magic Squares with 5-Digits (0,1,6,8,9)<\/mark><\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-8c93778b189769de2931b8c70dda159f\">Example 1: Blocks of Magic Squares of Order 4<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"2509\" height=\"1587\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01689a.png\" alt=\"\" class=\"wp-image-13187\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01689a.png 2509w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01689a-300x190.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01689a-1024x648.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01689a-768x486.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01689a-535x338.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01689a-1536x972.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01689a-2048x1295.png 2048w\" sizes=\"(max-width: 2509px) 100vw, 2509px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>upside-down<\/strong>&nbsp;with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(0,1,6,8,9):=11120109.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with <strong>different<\/strong> magic sums<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-2c82984ab7efadcfabde7a970407bffb\">Example 2: Blocks of Magic Squares of Order 5<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"2499\" height=\"1418\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-01689a.png\" alt=\"\" class=\"wp-image-13188\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-01689a.png 2499w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-01689a-300x170.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-01689a-1024x581.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-01689a-768x436.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-01689a-535x304.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-01689a-1536x872.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-01689a-2048x1162.png 2048w\" sizes=\"(max-width: 2499px) 100vw, 2499px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>upside-down<\/strong>&nbsp;with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(0,1,6,8,9):=11120109.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are <strong>pandiagonal<\/strong> magic squares with different magic sums.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">Universal Magic Squares with 5-Digits (0,1,2,5,8)<\/mark><\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-1a6edde3ce489c9893c883cf96cc9d5f\">Example 3: Blocks of Magic Squares of Order 4<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"2610\" height=\"1305\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01258a.png\" alt=\"\" class=\"wp-image-13191\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01258a.png 2610w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01258a-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01258a-1024x512.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01258a-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01258a-535x268.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01258a-1536x768.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-01258a-2048x1024.png 2048w\" sizes=\"(max-width: 2610px) 100vw, 2610px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>universal <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(0,1,2,5,8):=81103.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with <strong>different<\/strong> magic sums<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-9f5777a89abdcbec500c12e2d3b29d84\">Example 4: Blocks of Magic Squares of Order 5<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2479\" height=\"1389\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-01258.png\" alt=\"\" class=\"wp-image-13192\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-01258.png 2479w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-01258-300x168.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-01258-1024x574.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-01258-768x430.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-01258-535x300.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-01258-1536x861.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-01258-2048x1148.png 2048w\" sizes=\"(max-width: 2479px) 100vw, 2479px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>universal <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(0,1,2,5,8):=81103.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are <strong>pandiagonal<\/strong> magic squares with <strong>different<\/strong> magic sums.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-fc76ea2193f8773edb397f7f823a4610\">b) 6-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">Upside-Down Magic Squares with 3-Digits (1,6,9)<\/mark><\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-e6eabac88fe5217e94c04ac3279745e6\">Example 5: Blocks of Magic Squares of Order 4<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2836\" height=\"1014\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-169.png\" alt=\"\" class=\"wp-image-13197\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-169.png 2836w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-169-300x107.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-169-1024x366.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-169-768x275.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-169-535x191.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-169-1536x549.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-169-2048x732.png 2048w\" sizes=\"(max-width: 2836px) 100vw, 2836px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>upside-down <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(1,6,9):=12203191.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with <strong>different<\/strong> magic sums.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-aa05644f11f44e193446782d9afafb94\">Example 6: Blocks of Magic Squares of Order 5<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2958\" height=\"997\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-169-2.png\" alt=\"\" class=\"wp-image-13210\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-169-2.png 2958w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-169-2-300x101.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-169-2-1024x345.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-169-2-768x259.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-169-2-535x180.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-169-2-1536x518.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-169-2-2048x690.png 2048w\" sizes=\"(max-width: 2958px) 100vw, 2958px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>upside-down <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(1,6,9):=12203191.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are <strong>pandiagonal<\/strong> magic squares with <strong>different<\/strong> magic sums.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">Universal Magic Squares with 2-Digits (1,8)<\/mark><\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-7ee8f9cbab093e75f6b7d22ef29717f7\">Example 7: Blocks of Magic Squares of Order 4<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3038\" height=\"1180\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-258b-1.png\" alt=\"\" class=\"wp-image-13213\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-258b-1.png 3038w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-258b-1-300x117.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-258b-1-1024x398.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-258b-1-768x298.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-258b-1-535x208.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-258b-1-1536x597.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-258b-1-2048x795.png 2048w\" sizes=\"(max-width: 3038px) 100vw, 3038px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>universal <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(2,5,8):=10867857.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with <strong>different<\/strong> magic sums.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-af5160f8611dab2731ef71ca52a7001a\">Example 8: Blocks of Magic Squares of Order 5<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3001\" height=\"1161\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-258a-1.png\" alt=\"\" class=\"wp-image-13214\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-258a-1.png 3001w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-258a-1-300x116.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-258a-1-1024x396.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-258a-1-768x297.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-258a-1-535x207.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-258a-1-1536x594.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-258a-1-2048x792.png 2048w\" sizes=\"(max-width: 3001px) 100vw, 3001px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>universal <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(2,5,8):=10867857.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are <strong>pandiagonal <\/strong>magic squares with <strong>different<\/strong> magic sums.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-luminous-vivid-orange-color has-text-color has-link-color wp-elements-5eb87b6132f517873c5c3b9270b9cacd\">c) 10-Digits Cell Entries:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">Universal Magic Squares with 2-Digits (1,8)<\/mark><\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-26758294f193979ad82d74f554d07be8\">Example 9: Blocks of Magic Squares of Order 4<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3107\" height=\"917\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-18-1.png\" alt=\"\" class=\"wp-image-13218\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-18-1.png 3107w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-18-1-300x89.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-18-1-1024x302.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-18-1-768x227.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-18-1-535x158.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-18-1-1536x453.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-18-1-2048x604.png 2048w\" sizes=\"(max-width: 3107px) 100vw, 3107px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>universal <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(1,8):=99999999990.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are <strong>pandiagonal<\/strong> magic squares with equal magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>4\u00d74<\/sub>(1,8):=19999999998.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-379da671759fb85a2a2b9b2ecc11cb44\">Example 11: Blocks of Magic Squares of Order 5<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3104\" height=\"921\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-18-1.png\" alt=\"\" class=\"wp-image-13217\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-18-1.png 3104w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-18-1-300x89.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-18-1-1024x304.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-18-1-768x228.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-18-1-535x159.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-18-1-1536x456.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-18-1-2048x608.png 2048w\" sizes=\"(max-width: 3104px) 100vw, 3104px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>universal <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(1,8):=99999999990.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are <strong>pandiagonal<\/strong> magic squares with <strong>different<\/strong> magic sums.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">Universal Magic Squares with 2-Digits (2,5)<\/mark><\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-32f98ff3cbf4b93eb1cb080c6c720c43\">Example 12: Blocks of Magic Squares of Order 4<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2898\" height=\"879\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-25-1.png\" alt=\"\" class=\"wp-image-13221\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-25-1.png 2898w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-25-1-300x91.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-25-1-1024x311.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-25-1-768x233.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-25-1-535x162.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-25-1-1536x466.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-25-1-2048x621.png 2048w\" sizes=\"(max-width: 2898px) 100vw, 2898px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>universal <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(2,5):=77777777770.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are <strong>pandiagonal<\/strong> magic squares with equal magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>4\u00d74<\/sub>(2,5):=15555555554.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-296a3d12eedab698082c6e8fdb2fe4f6\">Example 10: Blocks of Magic Squares of Order 5<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2935\" height=\"916\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-25-1.png\" alt=\"\" class=\"wp-image-13222\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-25-1.png 2935w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-25-1-300x94.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-25-1-1024x320.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-25-1-768x240.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-25-1-535x167.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-25-1-1536x479.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-25-1-2048x639.png 2048w\" sizes=\"(max-width: 2935px) 100vw, 2935px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>universal <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(2,5):=77777777770.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are <strong>pandiagonal<\/strong> magic squares with <strong>different<\/strong> magic sums.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-red-color\">Upside-Down Magic Squares with 2-Digits (6,9)<\/mark><\/h3>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-4fe31c697726b8ad1bfe9a005d762bc5\">Example 13: Blocks of Magic Squares of Order 4<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3106\" height=\"877\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-69-1.png\" alt=\"\" class=\"wp-image-13223\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-69-1.png 3106w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-69-1-300x85.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-69-1-1024x289.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-69-1-768x217.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-69-1-535x151.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-69-1-1536x434.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-4x4-69-1-2048x578.png 2048w\" sizes=\"(max-width: 3106px) 100vw, 3106px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>upside-down <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(6,9):=166666666650.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are <strong>pandiagonal<\/strong> magic squares with equal magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>4\u00d74<\/sub>(6,9):=33333333330.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-ba461496c22ff02c304c5c34c311f8ad\">Example 14: Blocks of Magic Squares of Order 5<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3107\" height=\"915\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-69.png\" alt=\"\" class=\"wp-image-13224\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-69.png 3107w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-69-300x88.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-69-1024x302.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-69-768x226.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-69-535x158.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-69-1536x452.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/20x20-5x5-69-2048x603.png 2048w\" sizes=\"(max-width: 3107px) 100vw, 3107px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is&nbsp;<strong>upside-down <\/strong>with&nbsp;<strong>magic<\/strong>&nbsp;sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>20\u00d720<\/sub>(6,9):=166666666650.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are <strong>pandiagonal<\/strong> magic squares with <strong>different<\/strong> magic sums.<\/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>, November 05, 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 and bimagic squares of order 16. 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":13188,"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-13186","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\/10\/20x20-5x5-01689a.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13186","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=13186"}],"version-history":[{"count":16,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13186\/revisions"}],"predecessor-version":[{"id":13387,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13186\/revisions\/13387"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/13188"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13186"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13186"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}