{"id":13336,"date":"2024-10-30T13:17:13","date_gmt":"2024-10-30T16:17:13","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=13336"},"modified":"2026-05-19T18:18:07","modified_gmt":"2026-05-19T21:18:07","slug":"universal-and-upside-down-magic-and-bimagic-squares-of-order-25","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=13336","title":{"rendered":"Universal and Upside-Down Magic and Bimagic Squares of Order 25"},"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 based on blocks of magic squares of orders 3, 4, 6 and 8. In some cases the these blocks are <strong>pandiagonal<\/strong> magic squares. The blocks of order 3 are always <strong>semi-magic<\/strong> squares.<\/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<strong> 5-digits<\/strong>, such as, (0,1,6,8,9) and (0,1,2,5,8). 5-Digits combinations lead us to 4-digits cell entries. This part bring us <strong>upside-down<\/strong> and <strong>universal<\/strong> <strong>pandiagonal<\/strong> <strong>bimagic squares<\/strong> of order 25, where the blocks of order 5 are <strong>pandiagonal<\/strong> magic squares of equal  magic sums. <\/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 <strong>3-digits<\/strong>, such as, (1,6,9) and (2,5,8). 3-Digits combinations lead us to 6-digits cell entries. This part bring us <strong>upside-down<\/strong> and <strong>universal<\/strong> <strong>pandiagonal<\/strong> <strong>magic squares <\/strong>of order 25. In this case, still, we don&#8217;t have <strong>bimagic squares<\/strong>. The blocks of order 5 are <strong>pandiagonal<\/strong> magic squares with different magic sums.<\/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. This part bring us <strong>upside-down<\/strong> and <strong>universal<\/strong> <strong>pandiagonal<\/strong> <strong>magic squares <\/strong>of order 25, where the blocks of order 5 are <strong>pandiagonal<\/strong> magic squares of different magic sums. <\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>12-Digits Cell Entries<\/strong>\n<ul class=\"wp-block-list\">\n<li>In this case, the work is in<strong> 2-digits<\/strong>, such as, (1,8), (2,5) and (6,9). 2-Digits combinations lead us to 12-digits cell entries. This part bring us <strong>upside-down<\/strong> and <strong>universal<\/strong> <strong>pandiagonal<\/strong> <strong>bimagic squares<\/strong> of order 25, where the blocks of order 5 are <strong>pandiagonal<\/strong> magic squares of equal  magic sums. <\/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 of orders 3 to 25 except the orders 17, 18, 19, 22 and 23 see the reference list given at the end of this work. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">More details of this work are given in following online work:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Inder J. Taneja<\/strong>, Universal and Upside-Down Magic Bimagic Squares of Order 25, <strong>Zenodo<\/strong>, October 30, 2024, pp. 1-43, <a href=\"https:\/\/doi.org\/10.5281\/zenodo.14014851.\">https:\/\/doi.org\/10.5281\/zenodo.14014851.<\/a><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(42,222,179) 53%,rgb(155,81,224) 100%)\">Universal and Upside-Down <br>Magic and Bimagic Squares of Order 25<\/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 Square 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-de18c330f670be6cd46620d70db51ab7\">Example 1: Upside-Down Pandiagonal Bimagic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"2243\" height=\"1288\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01689.png\" alt=\"\" class=\"wp-image-13337\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01689.png 2243w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01689-300x172.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01689-1024x588.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01689-768x441.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01689-535x307.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01689-1536x882.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01689-2048x1176.png 2048w\" sizes=\"(max-width: 2243px) 100vw, 2243px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>upside-down<\/strong>&nbsp;<strong>pandiagonal bimagic<\/strong> square with&nbsp;<strong>magic<\/strong>&nbsp;and <strong>bimagic<\/strong> sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(0,1,6,8,9):=133320<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>Sb<sub>25\u00d725<\/sub>(0,1,6,8,9):=1048342630<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are pandiagonal magic squares with <strong>equal <\/strong>magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>5\u00d75<\/sub>(0,1,6,8,9):=26664<\/strong><\/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 Square 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-93f0c8aee6de36af6afec3b3a6362935\">Example 2: Universal  Pandiagonal Bimagic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"2520\" height=\"1289\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01258-1.png\" alt=\"\" class=\"wp-image-13339\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01258-1.png 2520w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01258-1-300x153.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01258-1-1024x524.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01258-1-768x393.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01258-1-535x274.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01258-1-1536x786.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-01258-1-2048x1048.png 2048w\" sizes=\"(max-width: 2520px) 100vw, 2520px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>universal pandiagonal bimagic<\/strong> square with&nbsp;<strong>magic<\/strong>&nbsp;and <strong>bimagic<\/strong> sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(0,1,6,8,9):=88880<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>Sb<sub>25\u00d725<\/sub>(0,1,6,8,9):=532147790<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are pandiagonal magic squares with <strong>equal <\/strong>magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>5\u00d75<\/sub>(0,1,6,8,9):=17776<\/strong>.<\/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-0d19e950acb21f31a75cd989c3c4c66f\">Example 3: Pandiagonal  Magic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"3191\" height=\"1221\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-169.png\" alt=\"\" class=\"wp-image-13340\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-169.png 3191w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-169-300x115.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-169-1024x392.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-169-768x294.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-169-535x205.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-169-1536x588.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-169-2048x784.png 2048w\" sizes=\"(max-width: 3191px) 100vw, 3191px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>upside-down<\/strong>&nbsp;<strong>pandiagonal<\/strong> magic square with&nbsp;magic sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(1,6,9):=14333319<\/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 3-Digits (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-6e866bc66fe08af87d4a02ff42f2bec0\">Example 4: Pandiagonal  Magic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3535\" height=\"1221\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-258.png\" alt=\"\" class=\"wp-image-13341\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-258.png 3535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-258-300x104.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-258-1024x354.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-258-768x265.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-258-535x185.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-258-1536x531.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-258-2048x707.png 2048w\" sizes=\"(max-width: 3535px) 100vw, 3535px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>universal pandiagonal<\/strong> magic square with&nbsp;magic sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(2,5,8):=14222208.<\/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<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-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-b3c526069d2067950c208b86e980f88c\">Example 5: Pandiagonal Magic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3237\" height=\"1234\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18a.png\" alt=\"\" class=\"wp-image-13342\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18a.png 3237w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18a-300x114.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18a-1024x390.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18a-768x293.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18a-535x204.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18a-1536x586.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18a-2048x781.png 2048w\" sizes=\"(max-width: 3237px) 100vw, 3237px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>universal pandiagonal<\/strong> magic square with&nbsp;magic sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(1,8):=128888888876<\/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 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-0c7151fa3d818c3a944b4b058b92ae2e\">Example 6: Pandiagonal Magic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3537\" height=\"1258\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25.png\" alt=\"\" class=\"wp-image-13343\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25.png 3537w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-300x107.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-1024x364.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-768x273.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-535x190.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-1536x546.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-2048x728.png 2048w\" sizes=\"(max-width: 3537px) 100vw, 3537px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>universal pandiagonal<\/strong> magic square with&nbsp;magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S1<sub>25\u00d725<\/sub>(2,5):=98888888879<\/strong> and <strong>S2<sub>25\u00d725<\/sub>(2,5):=95555555546<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are two magic sums. The first one is for the original and upside-down, while the second sum for the mirror looking case. 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\">Upside-Down Magic Square 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-dc1a58464e7d795a1b76f1d6923bea8a\">Example 7: Pandiagonal Magic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3068\" height=\"1121\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69.png\" alt=\"\" class=\"wp-image-13344\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69.png 3068w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-300x110.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-1024x374.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-768x281.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-535x195.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-1536x561.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-2048x748.png 2048w\" sizes=\"(max-width: 3068px) 100vw, 3068px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>upside-down pandiagonal<\/strong> magic square with&nbsp;magic sum:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(6,9):=98888888879.<\/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-luminous-vivid-orange-color has-text-color has-link-color wp-elements-24e9d10ad98260172d5f313e7680586c\">d) 12-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 Bimagic Square 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-f64a31b73d24a3533a043e9a60763a9b\">Example 8: Universal Pandiagonal Bimagic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3366\" height=\"889\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18-bi.png\" alt=\"\" class=\"wp-image-13347\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18-bi.png 3366w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18-bi-300x79.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18-bi-1024x270.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18-bi-768x203.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18-bi-535x141.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18-bi-1536x406.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-18-bi-2048x541.png 2048w\" sizes=\"(max-width: 3366px) 100vw, 3366px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>upside-down<\/strong>&nbsp;<strong>pandiagonal bimagic<\/strong> square with&nbsp;<strong>magic<\/strong>&nbsp;and <strong>bimagic<\/strong> sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(1,8):=14794790205195<\/strong>,<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>Sb<sub>25\u00d725<\/sub>(1,8):=11527169460366128144971255<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are pandiagonal magic squares with <strong>equal <\/strong>magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>5\u00d75<\/sub>(1,8):=2958958041039.<\/strong><\/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 Square 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-0cc858211c3d778fb388080e2ffecb7c\">Example 9: Universal Pandiagonal Bimagic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3589\" height=\"890\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-bi.png\" alt=\"\" class=\"wp-image-13348\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-bi.png 3589w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-bi-300x74.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-bi-1024x254.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-bi-768x190.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-bi-535x133.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-bi-1536x381.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-25-bi-2048x508.png 2048w\" sizes=\"(max-width: 3589px) 100vw, 3589px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>universal pandiagonal bimagic<\/strong> square with&nbsp;<strong>magic<\/strong>&nbsp;and <strong>bimagic<\/strong> sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(2,5):=10705703738730<\/strong>,<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>Sb<sub>25\u00d725<\/sub>(2,5):=5093578210752809098589610<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are pandiagonal magic squares with <strong>equal <\/strong>magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>5\u00d75<\/sub>(2,5):=2141140747746.<\/strong><\/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 Bimagic Square 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-e096e8591a73c73fb481aff853b331d1\">Example 10: Upside-Down Pandiagonal Bimagic Square of Order 25<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3587\" height=\"888\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-bi.png\" alt=\"\" class=\"wp-image-13349\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-bi.png 3587w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-bi-300x74.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-bi-1024x254.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-bi-768x190.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-bi-535x132.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-bi-1536x380.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25x25-5x5-69-bi-2048x507.png 2048w\" sizes=\"(max-width: 3587px) 100vw, 3587px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">It is <strong>upside-down<\/strong>&nbsp;<strong>pandiagonal bimagic<\/strong> square with&nbsp;<strong>magic<\/strong>&nbsp;and <strong>bimagic<\/strong> sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>25\u00d725<\/sub>(6,9):=21816814849830<\/strong>,<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>Sb<sub>25\u00d725<\/sub>(6,9):=19548030916765021312550250<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 5 are pandiagonal magic squares with <strong>equal <\/strong>magic sums:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>5\u00d75<\/sub>(6,9):=4363362969966<\/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\n\n\n<p class=\"has-pale-ocean-gradient-background has-background wp-block-paragraph\"><strong>Note:<\/strong><br>The result for the magic squares of orders 17, 18, 19, 22 and 23 are missing. The reason is that the numbers 17, 19 and 23 are prime numbers. It can be done similar to orders 11 and 13. The other numbers missing are of orders 18 and 22. It can be done similar to orders 10 or 14.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Higher Order Bimagic Squares<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Inder J. Taneja<\/strong>, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011, pp. 1-14, (22.02.2011), h<a href=\"https:\/\/doi.org\/10.48550\/arXiv.1102.3052\">ttps:\/\/doi.org\/10.48550\/arXiv.1102.3052<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11031\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares: Multiples of Orders 8 and 16<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11090\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares Multiples of 25: Orders 25, 125 and 625<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11120\">Block-Wise Construction of Bimagic Squares of Orders 49 and 343.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11129\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares Multiples of 9: Orders 9, 81 and 729<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11142\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares of Orders 121 and 1331<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11168\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 256, 512 and 1024: Blocks of Order 16.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11186\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 200 and 1000: Blocks of Order 8.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11198\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 400, 800, 1600 and 2000: Blocks of Order 16.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11224\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 100, 110 and 121: Blocks of Orders 10 and 11.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=13138\">Universal and Upside-Down Magic and Bimagic Squares of Order 16<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=13336\">Universal and Upside-Down Magic and Bimagic Squares of Order 25<\/a><\/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":13337,"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-13336","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\/25x25-5x5-01689.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13336","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=13336"}],"version-history":[{"count":8,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13336\/revisions"}],"predecessor-version":[{"id":18915,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13336\/revisions\/18915"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/13337"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13336"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13336"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}