{"id":13138,"date":"2024-10-16T17:28:55","date_gmt":"2024-10-16T20:28:55","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=13138"},"modified":"2026-05-19T18:16:42","modified_gmt":"2026-05-19T21:16:42","slug":"universal-and-upside-down-magic-and-bimagic-squares-of-order-16","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=13138","title":{"rendered":"Universal and Upside-Down Magic and Bimagic Squares of Order 16"},"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\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This work bring 11 examples in different categories of entries. These are divided in three parts. <\/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 4-digits, such as, (1,6,8,9), (1,2,5,8) and (0,2,5,8). Four 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). Three digits combinations lead us to 6-digits cell entries.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>8<strong>-Digits Cell Entries<\/strong>\n<ul class=\"wp-block-list\">\n<li>In this case, the work is in 2-digits, such as, (6,9), (2,5) and (2,5). Two digits combinations lead us to 4-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> see the reference list given at the end of this work. 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 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><\/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<br>Magic and Bimagic Squares of Order 16<\/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-433ac506aaeebdf9cf90bb1c050e32b3\">Example 1: Upside-Down Bimagic Square in 4-digits (1,6,8,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"1214\" height=\"839\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1689.png\" alt=\"\" class=\"wp-image-13141\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1689.png 1214w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1689-300x207.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1689-1024x708.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1689-768x531.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1689-535x370.png 535w\" sizes=\"(max-width: 1214px) 100vw, 1214px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>bimagic<\/strong> and <strong>upside-down<\/strong> with <strong>magic<\/strong> and <strong>bimagic<\/strong> sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"> <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(1,6,8,9):=106656<\/strong><\/mark> and <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Sb<sub>16&#215;16<\/sub>(1,6,8,9):=864504248<\/strong><\/mark>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e.,  <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(1,6,8,9):=26664<\/mark><\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-8b2a53c3a04f47b83fc88245ee52cd9f\">Example 2:  Universal Bimagic Square in 4-digits (1,2,5,8)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" width=\"1597\" height=\"840\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1258.png\" alt=\"\" class=\"wp-image-13142\" style=\"width:1182px;height:auto\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1258.png 1597w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1258-300x158.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1258-1024x539.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1258-768x404.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1258-535x281.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-1258-1536x808.png 1536w\" sizes=\"(max-width: 1597px) 100vw, 1597px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>bimagic<\/strong> and <strong>universal <\/strong>with <strong>magic<\/strong> and <strong>bimagic<\/strong> sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(1,2,5,8):=71104<\/strong><\/mark> and <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Sb<sub>16&#215;16<\/sub>(1,2,5,8):=437198296<\/strong><\/mark>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e., <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(1,2,5,8):=17776<\/mark><\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-45bb662e101133cd9ee3d57c4dc6d7de\">Example 3: Universal Bimagic Square in 4-digits (0,2,5,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"1490\" height=\"714\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-0258.png\" alt=\"\" class=\"wp-image-13143\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-0258.png 1490w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-0258-300x144.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-0258-1024x491.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-0258-768x368.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/16x16-0258-535x256.png 535w\" sizes=\"(max-width: 1490px) 100vw, 1490px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>bimagic<\/strong> and <strong>universal <\/strong>with <strong>magic<\/strong> and <strong>bimagic<\/strong> sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(0,2,5,8):=66660<\/strong><\/mark> and <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Sb<sub>16&#215;16<\/sub>(0,2,5,8):=  426207072<\/strong><\/mark>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e., <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(0,2,5,8):=16665<\/mark><\/strong>.<\/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-vivid-purple-color has-text-color has-link-color wp-elements-5ab0f31290b1a06a959db023607b08bf\">Example 4: Upside-Down Magic Square in 3-digits (1,6,9)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1539\" height=\"699\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-169.png\" alt=\"\" class=\"wp-image-12759\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-169.png 1539w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-169-300x136.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-169-1024x465.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-169-768x349.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-169-535x243.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-169-1536x698.png 1536w\" sizes=\"(max-width: 1539px) 100vw, 1539px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The above magic square in 3-digits (1,6,9) is <strong>upside-down<\/strong> with magic sum<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(2,5,8):=11120109<\/strong><\/mark>. <\/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-262aed2d42999fd69b9b69c5f9cacdba\">Example 5: Universal Magic Square in 3-digits (2,5,8)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1619\" height=\"633\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-258.png\" alt=\"\" class=\"wp-image-12760\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-258.png 1619w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-258-300x117.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-258-1024x400.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-258-768x300.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-258-535x209.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/15x15-258-1536x601.png 1536w\" sizes=\"(max-width: 1619px) 100vw, 1619px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square in 3-digits (2,5,8) is <strong>universal. <\/strong>In case of <strong>original<\/strong> and <strong>mirror looking<\/strong> versions, the magic sums are same, i.e, <strong> <\/strong> <\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>SO<sub>16&#215;16<\/sub>(2,5,8):=SM<sub>16&#215;16<\/sub>(2,5,8):=8005998<\/strong><\/mark>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In case of <strong>upside-down<\/strong>, the magic sum is different, i.e., <\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>SU<sub>16&#215;16<\/sub>(2,5,8):=7708701<\/strong><\/mark>. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The blocks of order 4 are 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-95d354e3411328dc37bd9ba55721356d\">b) 8-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-ad0a3b2d15939047967564d827101540\">Example 6. Upside-Down Bimagic Square in 2-digits (6,9)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3051\" height=\"975\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-bi.png\" alt=\"\" class=\"wp-image-13148\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-bi.png 3051w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-bi-300x96.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-bi-1024x327.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-bi-768x245.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-bi-535x171.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-bi-1536x491.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-bi-2048x654.png 2048w\" sizes=\"(max-width: 3051px) 100vw, 3051px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>bimagic<\/strong> and <strong>upside-down<\/strong> with <strong>magic<\/strong> and <strong>bimagic<\/strong> sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(6,9):=1333333320<\/strong><\/mark> and <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Sb<sub>16&#215;16<\/sub>(6,9):=114747472525252536<\/strong><\/mark>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e., <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(6,9):=333333330<\/mark><\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-0294f3af3ea31e819267fd3094bfe3aa\">Example 7. Upside-Down Pandiagonal Magic Square in 2-digits (6,9)<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3056\" height=\"974\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-pan.png\" alt=\"\" class=\"wp-image-13153\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-pan.png 3056w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-pan-300x96.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-pan-1024x326.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-pan-768x245.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-pan-535x171.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-pan-1536x490.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/69-16x16-pan-2048x653.png 2048w\" sizes=\"(max-width: 3056px) 100vw, 3056px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>upside-down<\/strong> and <strong>pandiagonal <\/strong>with magic sum<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(6,9):=1333333320<\/strong><\/mark><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e., <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(6,9):=333333330<\/mark><\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-f9cfcee99c61ca35fd2090d62645d1f4\">Example 8. Universal Bimagic Square in 2-digits (1,8)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2704\" height=\"895\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-bi.png\" alt=\"\" class=\"wp-image-13150\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-bi.png 2704w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-bi-300x99.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-bi-1024x339.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-bi-768x254.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-bi-535x177.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-bi-1536x508.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-bi-2048x678.png 2048w\" sizes=\"(max-width: 2704px) 100vw, 2704px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>bimagic<\/strong> and <strong>universal<\/strong> with <strong>magic<\/strong> and <strong>bimagic<\/strong> sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(1,8):=799999992<\/strong><\/mark> and <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Sb<sub>16&#215;16<\/sub>(1,8):=59797978997979800<\/strong><\/mark>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e., <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(1,8):=199999998<\/mark><\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-895c0f901065e43bb7ef569ecc4c79a9\">Example 9. Universal Pandiagonal Magic Square in 2-digits (1,8)<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2702\" height=\"891\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-pan.png\" alt=\"\" class=\"wp-image-13154\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-pan.png 2702w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-pan-300x99.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-pan-1024x338.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-pan-768x253.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-pan-535x176.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-pan-1536x507.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/18-16x16-pan-2048x675.png 2048w\" sizes=\"(max-width: 2702px) 100vw, 2702px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>universal<\/strong> and pandiagonal with magic sum<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(1,8):=799999992<\/strong><\/mark><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e., <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(1,8):=199999998<\/mark><\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-652869bc3f69a7b47d86f7a8a4b26029\">Example 10. Universal Bimagic Square in 2-digits (2,5)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2819\" height=\"972\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-bi.png\" alt=\"\" class=\"wp-image-13151\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-bi.png 2819w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-bi-300x103.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-bi-1024x353.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-bi-768x265.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-bi-535x184.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-bi-1536x530.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-bi-2048x706.png 2048w\" sizes=\"(max-width: 2819px) 100vw, 2819px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>bimagic<\/strong> and <strong>universal<\/strong> with <strong>magic<\/strong> and <strong>bimagic<\/strong> sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(2,5):=622222216<\/strong><\/mark> and <mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>Sb<sub>16&#215;16<\/sub>(2,5):=27833894016610552<\/strong><\/mark>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e., <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(2,5):=155555554<\/mark><\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-c6cd7cb072aab85a41e64a60e8de58f6\">Example 11. Universal Pandiagonal Magic Square in 2-digits (2,5)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2818\" height=\"974\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-pan.png\" alt=\"\" class=\"wp-image-13155\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-pan.png 2818w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-pan-300x104.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-pan-1024x354.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-pan-768x265.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-pan-535x185.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-pan-1536x531.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2024\/10\/25-16x16-pan-2048x708.png 2048w\" sizes=\"(max-width: 2818px) 100vw, 2818px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">The above magic square is <strong>universal<\/strong> and <strong>pandiagonal<\/strong> with magic sum<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\"><strong>S<sub>16&#215;16<\/sub>(2,5):=622222216<\/strong><\/mark>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Blocks of order 4 are magic squares with equal magic sums, i.e., <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-purple-color\">S<sub>4&#215;4<\/sub>(2,5):=155555554<\/mark><\/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>, 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\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":13141,"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-13138","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\/16x16-1689.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13138","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=13138"}],"version-history":[{"count":14,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13138\/revisions"}],"predecessor-version":[{"id":18914,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/13138\/revisions\/18914"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/13141"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}