{"id":11031,"date":"2023-12-12T10:54:37","date_gmt":"2023-12-12T13:54:37","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11031"},"modified":"2023-12-13T12:19:35","modified_gmt":"2023-12-13T15:19:35","slug":"block-wise-construction-of-bimagic-squares-multiples-of-orders-8-and-16","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=11031","title":{"rendered":"Block-Wise Construction of Bimagic Squares: Multiples of Orders 8 and 16"},"content":{"rendered":"\n<p class=\"has-text-align-right\"><strong><em>Whole the work is done manually on excel sheets<\/em><\/strong>.<\/p>\n\n\n\n<p class=\"has-text-align-justify has-electric-grass-gradient-background has-background\">Below are <strong>bimagic squares<\/strong> written in blocks multiples of orders 8 and 16. Most of the work is done manually by author in 2011. Few of them are with the help of <strong>Aale de Winkel<\/strong> &lt;aaledewinkel@magichypercubes.com&gt;, <a href=\"https:\/\/www.magichypercubes.com\/Encyclopedia\/\">Magic Encyclopedia magic square, magic cube, magic tesseract, magic hypercube<\/a>. See the reference below:<\/p>\n\n\n\n<ul 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<\/ul>\n\n\n\n<p>Before proceeding further below are the basic formulas to check the sums of <strong>magic<\/strong> and <strong>bimagic<\/strong> squares<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Magic Sum<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"176\" height=\"88\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/S-5.png\" alt=\"\" class=\"wp-image-11160\"\/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Bimagic Sum<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"304\" height=\"93\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-5.png\" alt=\"\" class=\"wp-image-11161\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-5.png 304w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-5-300x92.png 300w\" sizes=\"(max-width: 304px) 100vw, 304px\" \/><\/figure>\n<\/div>\n\n\n<p>where <strong><em>n<\/em><\/strong> is the order the magic square.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>8&#215;8<\/sub>:=260; Sb<sub>8&#215;8<\/sub>:= 11180<\/strong><\/p>\n\n\n\n<p>2&#215;4 blocks are of equal sums as of magic square, i.e., 260. See below the <strong>bimagic square<\/strong> of order 8.<\/p>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"780\" height=\"433\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1.png\" alt=\"\" class=\"wp-image-11079\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1.png 780w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1-300x167.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1-768x426.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/8X8-bi-1-535x297.png 535w\" sizes=\"(max-width: 780px) 100vw, 780px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-justify\"><em>The construction of <\/em>bimagic<em> square of order 8 is well known in the history, and is done by G. Pfeffermann in 1891. In this case, we have a <strong>pandiagonal bimagic<\/strong> square of order 8, where the blocks of order 2&#215;4 are of same sum as of magic square of order 8.<\/em><\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 16<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>16&#215;16<\/sub>:=2056; Sb<sub>16&#215;16<\/sub>:= 351576, S<sub>4&#215;4<\/sub>:=(1\/4) S<sub>16&#215;16<\/sub>=514.<\/strong><\/p>\n\n\n\n<p>See below the <strong>bimagic square<\/strong> of order 16.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1700\" height=\"758\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2.png\" alt=\"\" class=\"wp-image-11106\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2.png 1700w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-300x134.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-1024x457.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-768x342.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-535x239.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi-2-1536x685.png 1536w\" sizes=\"(max-width: 1700px) 100vw, 1700px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Semi-Bimagic Square of Order 24: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>24&#215;24<\/sub>:=6924; Sb<sub>24&#215;24<\/sub>:= 2661124 (semi), S<sub>8&#215;8<\/sub>:=2308.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either bimagic or semi-bimagic. See semi-bimagic square of order 24<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2729\" height=\"1035\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/24x24-semi-bi-1.png\" alt=\"\" class=\"wp-image-11107\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/24x24-semi-bi-1.png 2729w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/24x24-semi-bi-1-300x114.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/24x24-semi-bi-1-1024x388.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/24x24-semi-bi-1-768x291.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/24x24-semi-bi-1-535x203.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/24x24-semi-bi-1-1536x583.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/24x24-semi-bi-1-2048x777.png 2048w\" sizes=\"(max-width: 2729px) 100vw, 2729px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 32: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>32&#215;32<\/sub>:=16400; Sb<sub>32&#215;32<\/sub>:= 11201200, S<sub>8&#215;8<\/sub>:=4100.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either bimagic or semi-bimagic. See <strong>bimagic square<\/strong> of order 32. It is also pandiagonal.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2017\" height=\"938\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/32X32a-bi.png\" alt=\"\" class=\"wp-image-11066\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/32X32a-bi.png 2017w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/32X32a-bi-300x140.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/32X32a-bi-1024x476.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/32X32a-bi-768x357.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/32X32a-bi-535x249.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/32X32a-bi-1536x714.png 1536w\" sizes=\"(max-width: 2017px) 100vw, 2017px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 40: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>40&#215;40<\/sub>:=32020; Sb<sub>40&#215;40<\/sub>:= 34165340, S<sub>8&#215;8<\/sub>:=6404.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify\"> Magic squares of order 8 are of equal sums. These are either bimagic or semi-bimagic. See <strong>bimagic square<\/strong> of order 40. It is also pandiagonal.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2701\" height=\"1329\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi.png\" alt=\"\" class=\"wp-image-11064\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi.png 2701w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-300x148.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-1024x504.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-768x378.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-535x263.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-1536x756.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/40X40-bi-2048x1008.png 2048w\" sizes=\"(max-width: 2701px) 100vw, 2701px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Semi-Bimagic Square of Order 48: Blocks of Order 16<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>48&#215;48<\/sub>:=55320; Sb<sub>48&#215;48<\/sub>:= 84989960 (semi-bimagic); S<sub>16&#215;16<\/sub>:=18440.<\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 16 are of equal sums. These are all <strong>bimagic<\/strong>. See below <strong>semi-bimagic square<\/strong> of order 48. <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3464\" height=\"1724\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/48X48-semi-bi.png\" alt=\"\" class=\"wp-image-11067\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/48X48-semi-bi.png 3464w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/48X48-semi-bi-300x149.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/48X48-semi-bi-1024x510.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/48X48-semi-bi-768x382.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/48X48-semi-bi-535x266.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/48X48-semi-bi-1536x764.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/48X48-semi-bi-2048x1019.png 2048w\" sizes=\"(max-width: 3464px) 100vw, 3464px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 56: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>56&#215;56<\/sub>:=87836; Sb<sub>56&#215;56<\/sub>:= 183665076; S<sub>8&#215;8<\/sub>:=12548.<\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either <strong>bimagic <\/strong>or<strong> semi-bimagic<\/strong> squares of order 8. See below <strong>bimagic square<\/strong> of order 56. <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2448\" height=\"1461\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/56X56-bi.png\" alt=\"\" class=\"wp-image-11071\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/56X56-bi.png 2448w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/56X56-bi-300x179.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/56X56-bi-1024x611.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/56X56-bi-768x458.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/56X56-bi-535x319.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/56X56-bi-1536x917.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/56X56-bi-2048x1222.png 2048w\" sizes=\"(max-width: 2448px) 100vw, 2448px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Squares of Order 64: Blocks of Orders 8 and 16<\/h3>\n\n\n\n<p>Below are two types of bimagic squares of order 64:<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">First Type: Blocks of Order 8<\/h4>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>64&#215;64<\/sub>:=131104; Sb<sub>64&#215;64<\/sub>:= 358045024; <strong>S<sub>8&#215;8<\/sub>:=16338<\/strong>.<\/strong>  <\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>See below <strong>bimagic square<\/strong> of order 64 with blocks of order 8. These blocks of order 8 are either bimagic or <strong>semi-bimagic<\/strong> with different sums, but the magic sum of all order magic squares is same. <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3241\" height=\"1660\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-8x8-1.png\" alt=\"\" class=\"wp-image-11074\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-8x8-1.png 3241w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-8x8-1-300x154.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-8x8-1-1024x524.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-8x8-1-768x393.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-8x8-1-535x274.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-8x8-1-1536x787.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-8x8-1-2048x1049.png 2048w\" sizes=\"(max-width: 3241px) 100vw, 3241px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Second Type: Blocks of Order 16<\/h4>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>64&#215;64<\/sub>:=131104; Sb<sub>64&#215;64<\/sub>:= 358045024; S<sub>16&#215;16<\/sub>:=32776 <strong>and <strong>S<sub>4&#215;4<\/sub>:==8194<\/strong><\/strong>.<\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 16 are of equal sums. These are all <strong>bimagic<\/strong>. See below <strong>bimagic square<\/strong> of order 64 with blocks of order 16. These blocks of order 16 are bimagic squares with different sums, but the magic sum of all order 16 magic squares is same. <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2854\" height=\"1731\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1.png\" alt=\"\" class=\"wp-image-11075\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1.png 2854w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1-300x182.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1-1024x621.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1-768x466.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1-535x324.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1-1536x932.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/64x64-16x16-1-2048x1242.png 2048w\" sizes=\"(max-width: 2854px) 100vw, 2854px\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 72: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>72&#215;72<\/sub>:=186660; Sb<sub>72&#215;72<\/sub>:=  645159180; S<sub>8&#215;8<\/sub>:=20740.<\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either bimagic ou semi-bimagic squares of different sums. See the excel sheet given at the end.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Squares of Order 80: Blocks of Orders 8 and 16<\/h3>\n\n\n\n<p>Similar to order 64, here also we have two ways to write bimagic square of order 80.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">First Type: Blocks of Order 8<\/h4>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>80&#215;80<\/sub>:=256040; <strong>Sb<sub>80&#215;80<\/sub><\/strong>:= 1092522680; <strong>S<sub>8&#215;8<\/sub>:=25604<\/strong>.<\/strong>   <\/p>\n\n\n\n<p class=\"has-text-align-justify\">These blocks of order 8 are either bimagic or <strong>semi-bimagic<\/strong> with different sums, but the magic sum of all order magic squares is same. See the magic square of order 80 in excel sheet given at the end.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Second Type: Blocks of Order 16<\/h4>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>80&#215;80<\/sub>:=256040; <strong>Sb<sub>80&#215;80<\/sub><\/strong>:= 1092522680; <strong>S<sub>16&#215;16<\/sub>:=<strong>51208<\/strong><\/strong> and <strong><strong>S<sub>4&#215;4<\/sub>:=<\/strong><\/strong>12802.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 16 are of equal sums. See below <strong>bimagic square<\/strong> of order 80 with blocks of order 16. These blocks of order 16 are <strong>bimagic squares<\/strong> with different bimagic sums, but the magic sum of all order 16 magic squares is same. See the magic square of order 80 in excel sheet given at the end.<\/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%)\">Bimagic Square of Order 88: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>88&#215;88<\/sub>:=340780; Sb<sub>88&#215;88<\/sub>:= 1759447140; S<sub>8&#215;8<\/sub>:=30980.<\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either <strong>bimagic<\/strong> ou <strong>semi-bimagic<\/strong> squares of different sums. See the excel sheet given at the end.<\/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%)\">Bimagic Square of Order 96: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>96&#215;96<\/sub>:=442416; Sb<sub>96&#215;96<\/sub>:= 2718351376; S<sub>8&#215;8<\/sub>:=36868.<\/strong>   <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either <strong>bimagic<\/strong> ou <strong>semi-bimagic<\/strong> squares of different sums. See the excel sheet given at the end. <br><br><strong><em>Even though 96 is divisible by 16, still, we don&#8217;t have bimagic square of order 96 with blocks of order 16<\/em><\/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%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 104: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>104&#215;104<\/sub>:=562484; Sb<sub>104&#215;104<\/sub>:=4056072124; S<sub>8&#215;8<\/sub>:=43268.<\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either <strong>bimagic<\/strong> ou <strong>semi-bimagic<\/strong> squares of different sums. See the excel sheet given at the end.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Squares of Order 112: Blocks of Orders 8 and 16<\/h3>\n\n\n\n<p> Similar to orders 64 and 80, here also we have two ways to writing <strong>bimagic square<\/strong> of order 112.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">First Type: Blocks of Order 8<\/h4>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>112&#215;112<\/sub>:=702520; <strong>Sb<sub>112&#215;112<\/sub><\/strong>:= 5875174760; <strong>S<sub>8&#215;8<\/sub>:=50180<\/strong>.<\/strong>   <\/p>\n\n\n\n<p class=\"has-text-align-justify\">These blocks of order 8 are either <strong>bimagic<\/strong> or <strong>semi-bimagic<\/strong> with different sums, but the magic sum of all order magic squares is same. See the magic square of order 112 in excel sheet given at the end.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Second Type: Blocks of Order 16<\/h4>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>112&#215;112<\/sub>:=702520; <strong>Sb<sub>112&#215;112<\/sub><\/strong>:= 5875174760; <strong>S<sub>16&#215;16<\/sub>:=100360<\/strong> and <strong><strong>S<sub>4&#215;4<\/sub>:=<\/strong><\/strong>25090.<\/strong>   <\/p>\n\n\n\n<p class=\"has-text-align-justify\">See below <strong>bimagic square<\/strong> of order 80 with blocks of order 16. These blocks of order 16 are <strong>bimagic squares<\/strong> with different bimagic sums, but the magic sums of all order 16, and order 4 are of equal 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%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 120: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>120&#215;120<\/sub>:=864060; Sb<sub>120&#215;120<\/sub>:=8295264020; S<sub>8&#215;8<\/sub>:=57604.<\/strong>  <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either <strong>bimagic<\/strong> ou <strong>semi-bimagic<\/strong> squares of different sums. See the excel sheet given at the end.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 128: Blocks of Orders 8 and 16<\/h3>\n\n\n\n<p>Similar to orders 64, 80 and 112, here also we have two ways to writing <strong>bimagic square<\/strong> of order 128.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">First Type: Blocks of Order 8<\/h4>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>128&#215;128<\/sub>:=1048640; <strong>Sb<sub>128&#215;128<\/sub><\/strong>:=11454294720; <strong>S<sub>8&#215;8<\/sub>:=65540<\/strong>.<\/strong>    <\/p>\n\n\n\n<p class=\"has-text-align-justify\">These blocks of order 8 are either <strong>bimagic<\/strong> or <strong>semi-bimagic<\/strong> with different sums, but the magic sum of all order magic squares is same. See the magic square of order 128 in excel sheet given at the end.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Second Type: Blocks of Order 16<\/h4>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>128&#215;128<\/sub>:=1048640; <strong>Sb<sub>128&#215;128<\/sub><\/strong>:=11454294720;<\/strong> <strong><strong>S<sub>16&#215;16<\/sub>:=131080<\/strong> and <strong><strong>S<sub>4&#215;4<\/sub>:=<\/strong><\/strong>32770.<\/strong>   <\/p>\n\n\n\n<p class=\"has-text-align-justify\">See below <strong>bimagic square<\/strong> of order 128 with blocks of order 16. These blocks of order 16 are <strong>bimagic squares<\/strong> with different bimagic sums, but the magic sums of all order 16, and order 4 are of equal sums. <\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 136: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>136&#215;136<\/sub>:=1257796; Sb<sub>136&#215;136<\/sub>:=15509882476; S<sub>8&#215;8<\/sub>:=73988.<\/strong>   <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either <strong>bimagic<\/strong> ou <strong>semi-bimagic<\/strong> squares of different sums. See the excel sheet given at the end. <\/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%)\">Bimagic Square of Order 144: Blocks of Order 16<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>144&#215;144<\/sub>:=1493064; <strong>Sb<sub>144&#215;144<\/sub><\/strong>:=20640614424;<\/strong> <strong><strong>S<sub>16&#215;16<\/sub>:=165896<\/strong> and <strong><strong>S<sub>4&#215;4<\/sub>:=<\/strong><\/strong>41474.<\/strong>   <\/p>\n\n\n\n<p class=\"has-text-align-justify\">See below <strong>bimagic square<\/strong> of order 128 with blocks of order 16. These blocks of order 16 are <strong>bimagic squares<\/strong> with different bimagic sums, but the magic sums of all order 16, and order 4 are of equal sums. <\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 152: Blocks of Order 8<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>152&#215;152<\/sub>:=1755980; Sb<sub>152&#215;152<\/sub>:=27047359940; S<sub>8&#215;8<\/sub>:=92420.<\/strong>   <\/p>\n\n\n\n<p class=\"has-text-align-justify\">Magic squares of order 8 are of equal sums. These are either <strong>bimagic<\/strong> ou <strong>semi-bimagic<\/strong> squares of different sums. See the excel sheet given at the end. <\/p>\n\n\n\n<p> <\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 160: Blocks of Order 16<\/h3>\n\n\n\n<p>In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>S<sub>144&#215;144<\/sub>:=2048080; <strong>Sb<sub>144&#215;144<\/sub><\/strong>:=34954581360;<\/strong> <strong><strong>S<sub>16&#215;16<\/sub>:=<strong>204808<\/strong><\/strong> and <strong><strong>S<sub>4&#215;4<\/sub>:=<\/strong><\/strong>51202.<\/strong>   <\/p>\n\n\n\n<p>See below <strong>bimagic square<\/strong> of order 128 with blocks of order 16. These blocks of order 16 are <strong>bimagic squares<\/strong> with different bimagic sums, but the magic sums of all order 16, and order 4 are of equal sums. <\/p>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Excel file of Bimagic Squares Multiples of Orders 8 and 16.<\/h3>\n\n\n\n<p><\/p>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-881d48e2-110a-494e-a628-5bb7fc418661\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Bimagic-Multiples-8-and-16.xlsx\">Bimagic-Multiples-8-and-16<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Bimagic-Multiples-8-and-16.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-881d48e2-110a-494e-a628-5bb7fc418661\">Download<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Whole the work is done manually on excel sheets. Below are bimagic squares written in blocks multiples of orders 8 and 16. Most of the work is done manually by author in 2011. Few of them are with the help of Aale de Winkel &lt;aaledewinkel@magichypercubes.com&gt;, Magic Encyclopedia magic square, magic cube, magic tesseract, magic hypercube. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":11060,"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-11031","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-magic-squares"],"jetpack_featured_media_url":"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/16x16-bi.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11031","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=11031"}],"version-history":[{"count":24,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11031\/revisions"}],"predecessor-version":[{"id":11162,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11031\/revisions\/11162"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/11060"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}