{"id":10686,"date":"2023-10-28T01:50:32","date_gmt":"2023-10-28T04:50:32","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=10686"},"modified":"2023-10-28T01:50:33","modified_gmt":"2023-10-28T04:50:33","slug":"pandiagonal-magic-squares-among-orders-4-to-108","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=10686","title":{"rendered":"Pandiagonal Magic Squares Among Orders 4 to 108"},"content":{"rendered":"\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(255,206,236) 0%,rgb(255,255,255) 59%,rgb(152,150,240) 100%)\">This work brings <strong>pandiagonal<\/strong> magic squares in different situations. These are multiples of order 3, 4, 5, etc. We can always write <strong>pandiagonal<\/strong> magic squares multiples of any order, except of type <strong>4k+2<\/strong>, where k=1, 2, 3, &#8230; etc. These orders are 6, 10, 14, 18, etc. Below are magic squares of orders 3, 4, 5 and 7:<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"533\" height=\"215\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-3-4.png\" alt=\"\" class=\"wp-image-10750\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-3-4.png 533w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-3-4-300x121.png 300w\" sizes=\"(max-width: 533px) 100vw, 533px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"762\" height=\"319\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-5-7.png\" alt=\"\" class=\"wp-image-10751\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-5-7.png 762w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-5-7-300x126.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-5-7-535x224.png 535w\" sizes=\"(max-width: 762px) 100vw, 762px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(255,206,236) 0%,rgb(255,255,255) 59%,rgb(152,150,240) 100%)\">We observe that except order 3 other magic squares are <strong>pandiagonal<\/strong>. Except order 4, the orders 5 and 7 are <strong>prime numbers<\/strong>. Starting from order 5, we can always write <strong>pandiagonal<\/strong> magic squares of <strong>prime orders<\/strong>. See below examples prime orders up to order 23:<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"699\" height=\"457\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-11.png\" alt=\"\" class=\"wp-image-10752\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-11.png 699w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-11-300x196.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-11-535x350.png 535w\" sizes=\"(max-width: 699px) 100vw, 699px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1007\" height=\"526\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-13.png\" alt=\"\" class=\"wp-image-10753\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-13.png 1007w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-13-300x157.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-13-768x401.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-13-535x279.png 535w\" sizes=\"(max-width: 1007px) 100vw, 1007px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1095\" height=\"649\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-17-1.png\" alt=\"\" class=\"wp-image-10707\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-17-1.png 1095w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-17-1-300x178.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-17-1-1024x607.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-17-1-768x455.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-17-1-535x317.png 535w\" sizes=\"(max-width: 1095px) 100vw, 1095px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1387\" height=\"741\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-19-2.png\" alt=\"\" class=\"wp-image-10694\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-19-2.png 1387w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-19-2-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-19-2-1024x547.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-19-2-768x410.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-19-2-535x286.png 535w\" sizes=\"(max-width: 1387px) 100vw, 1387px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1438\" height=\"851\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-23.png\" alt=\"\" class=\"wp-image-10692\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-23.png 1438w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-23-300x178.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-23-1024x606.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-23-768x454.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/p-23-535x317.png 535w\" sizes=\"(max-width: 1438px) 100vw, 1438px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(255,206,236) 0%,rgb(255,255,255) 59%,rgb(152,150,240) 100%)\">Further orders such as 29, 31, 37, 41, etc. can also be written as pandiagonal magic squares with the same procedure as above. <\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 53%,rgb(0,208,130) 100%)\">Below are pandiagonal magic squares of non prime orders, such as 8, 12, 15, 16, 20, 21, etc. In these cases, we have pandiagonal magic squares with blocks of orders 3, 4, 5, 7, 11, 13, etc. These blocks are of two types. Either <strong>equal sums<\/strong> or <strong>non-equal sums<\/strong>. <br><br>For construction details the readers see the following author&#8217;s work. In some cases, magic rectangles are used to bring these pandiagonal magic squares:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"has-background wp-block-list\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgb(255,255,255) 49%,rgb(155,81,224) 100%)\">\n<li>Inder J. Taneja, Block-Wise Equal Sums Pandiagonal Magic Squares of Order 4k,&nbsp;<strong>Zenodo<\/strong>, January 31, 2019, pp. 1-17,&nbsp;<a href=\"https:\/\/doi.org\/10.5281\/zenodo.2554288\">https:\/\/doi.org\/10.5281\/zenodo.2554288<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>, Magic Rectangles in Construction of Block-Wise Pandiagonal Magic Squares,&nbsp;<strong>Zenodo<\/strong>, January 31, 2019, pp. 1-49,&nbsp;<a href=\"http:\/\/doi.org\/10.5281\/zenodo.2554520\">http:\/\/doi.org\/10.5281\/zenodo.2554520<\/a>.<\/li>\n\n\n\n<li>Inder Jeet Taneja, Block-Wise and Block-Bordered Magic and Bimagic Squares of Orders 10 to 47.&nbsp;<strong>Zenodo<\/strong>, January 14, 2021, pp. 1-185,&nbsp;<a href=\"http:\/\/doi.org\/10.5281\/zenodo.4437783\">http:\/\/doi.org\/10.5281\/zenodo.4437783<\/a>.<\/li>\n\n\n\n<li>Inder Jeet Taneja, Block-Wise Bordered and Pandiagonal Magic Squares Multiples of 4,&nbsp;<strong>Zenodo<\/strong>, August 31, 2021, pp. 1-148,&nbsp;<a href=\"https:\/\/doi.org\/10.5281\/zenodo.5347897\">https:\/\/doi.org\/10.5281\/zenodo.5347897<\/a>.<\/li>\n\n\n\n<li>Inder J. Taneja, A Simplified Procedure to Construct Pandiagonal Magic Squares Multiples of 4 (Revised),&nbsp;<strong>Zenodo<\/strong>, August 10, 2023, pp. 1-27,&nbsp;<a href=\"https:\/\/doi.org\/10.5281\/zenodo.8236754\">https:\/\/doi.org\/10.5281\/zenodo.8236754<\/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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Square of Order 8<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">Below is a pandiagonal magic square of order 8, constructed with equal sums of pandiagonal magic squares of order 4.<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"502\" height=\"353\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-8x8-4x4-1.png\" alt=\"\" class=\"wp-image-10703\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-8x8-4x4-1.png 502w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-8x8-4x4-1-300x211.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-8x8-4x4-1-500x353.png 500w\" sizes=\"(max-width: 502px) 100vw, 502px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Square of Order 9<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">Below is a pandiagonal magic square of order 9, constructed with blocks of equal sums of semi-magic squares of order 3.<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"556\" height=\"416\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-9x9-3x3-1.png\" alt=\"\" class=\"wp-image-10768\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-9x9-3x3-1.png 556w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-9x9-3x3-1-300x224.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-9x9-3x3-1-535x400.png 535w\" sizes=\"(max-width: 556px) 100vw, 556px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Square of Order 12<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 12=3&#215;4. In this we have pandiagonal magic squares of order 12 with blocks of order 3 and 4.  The first magic squares of order 12 is with different sums magic squares of order 3. The second magic square of order 12 is with equal sums pandiagonal magic squares of order 4. See below:<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"726\" height=\"493\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-12x12-3x3-2.png\" alt=\"\" class=\"wp-image-10701\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-12x12-3x3-2.png 726w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-12x12-3x3-2-300x204.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-12x12-3x3-2-535x363.png 535w\" sizes=\"(max-width: 726px) 100vw, 726px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"722\" height=\"489\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-12x12-4x4-3.png\" alt=\"\" class=\"wp-image-10702\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-12x12-4x4-3.png 722w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-12x12-4x4-3-300x203.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-12x12-4x4-3-535x362.png 535w\" sizes=\"(max-width: 722px) 100vw, 722px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 15<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 15=3&#215;5. In this we have pandiagonal magic squares of order 15 with blocks of order 3 and 5.  The first magic squares of order 15 is with equal sums semi-magic squares of squares of order 3. The second magic square of order 15 is with equal sums pandiagonal magic squares of order 5. See below:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1143\" height=\"598\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-3x3-1.png\" alt=\"\" class=\"wp-image-10704\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-3x3-1.png 1143w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-3x3-1-300x157.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-3x3-1-1024x536.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-3x3-1-768x402.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-3x3-1-535x280.png 535w\" sizes=\"(max-width: 1143px) 100vw, 1143px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1139\" height=\"596\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-5x5-1.png\" alt=\"\" class=\"wp-image-10705\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-5x5-1.png 1139w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-5x5-1-300x157.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-5x5-1-1024x536.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-5x5-1-768x402.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-15x15-5x5-1-535x280.png 535w\" sizes=\"(max-width: 1139px) 100vw, 1139px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(255,206,236) 0%,rgb(255,255,255) 59%,rgb(152,150,240) 100%)\"> We observe that the further orders in each alternative case have equal sums blocks. For blocks of order 3, the equal sums are for the magic squares of order 21, 27, 33, etc. In this case, we always have semi-magic squares of order 3. For blocks of order 5, the equal sums are for orders, 25, 35, 45, etc.<\/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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Square of Order 16<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\"> Below is a pandiagonal magic square of order 16, constructed with equal sums pandiagonal magic squares of order 4.<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1210\" height=\"689\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-16x16-4x4-1.png\" alt=\"\" class=\"wp-image-10706\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-16x16-4x4-1.png 1210w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-16x16-4x4-1-300x171.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-16x16-4x4-1-1024x583.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-16x16-4x4-1-768x437.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-16x16-4x4-1-535x305.png 535w\" sizes=\"(max-width: 1210px) 100vw, 1210px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 20<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 20=4&#215;5. In this we have pandiagonal magic squares with blocks of orders 4 and 5. The first pandiagonal magic square of order 20 is with equal sums pandiagonal magic squares of order 4. The second pandiagonal magic squares is with different sums pandiagonal magic squares of order 5. <\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1356\" height=\"859\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-4x4-1.png\" alt=\"\" class=\"wp-image-10709\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-4x4-1.png 1356w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-4x4-1-300x190.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-4x4-1-1024x649.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-4x4-1-768x487.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-4x4-1-535x339.png 535w\" sizes=\"(max-width: 1356px) 100vw, 1356px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1359\" height=\"859\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-5x5-1.png\" alt=\"\" class=\"wp-image-10710\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-5x5-1.png 1359w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-5x5-1-300x190.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-5x5-1-1024x647.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-5x5-1-768x485.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-20x20-5x5-1-535x338.png 535w\" sizes=\"(max-width: 1359px) 100vw, 1359px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 21<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 21=3&#215;7. In this we have pandiagonal magic squares with blocks of orders 3 and 7. The first magic square of order 21 is with equal sums semi-magic squares of order 3. The second magic square of order 21 is with equal sums pandiagonal magic squares of order 7. See below<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1387\" height=\"740\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-3x3-1.png\" alt=\"\" class=\"wp-image-10711\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-3x3-1.png 1387w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-3x3-1-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-3x3-1-1024x546.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-3x3-1-768x410.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-3x3-1-535x285.png 535w\" sizes=\"(max-width: 1387px) 100vw, 1387px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1382\" height=\"739\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-7x7-1.png\" alt=\"\" class=\"wp-image-10712\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-7x7-1.png 1382w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-7x7-1-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-7x7-1-1024x548.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-7x7-1-768x411.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-21x21-7x7-1-535x286.png 535w\" sizes=\"(max-width: 1382px) 100vw, 1382px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(255,206,236) 0%,rgb(255,255,255) 59%,rgb(152,150,240) 100%)\">We observe that the further orders in each alternative case, for order 7, we have equal sums blocks. For examples, magic squares of orders 35, 49, 63, etc. are with equal sums pandiagonal blocks of order 7<\/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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 24<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 24=2x3x4. In this we have pandiagonal magic squares with blocks of orders 3 and 4. The first magic square is made with 4 equal sums pandiagonal magic square of order 12. Each block of order 12 is formed by unequal sums of magic squares of order 3. The second magic square is made with equal sums pandiagonal magic squares of order 4.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1518\" height=\"834\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-12x12-3x3-2.png\" alt=\"\" class=\"wp-image-10725\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-12x12-3x3-2.png 1518w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-12x12-3x3-2-300x165.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-12x12-3x3-2-1024x563.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-12x12-3x3-2-768x422.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-12x12-3x3-2-535x294.png 535w\" sizes=\"(max-width: 1518px) 100vw, 1518px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1523\" height=\"833\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-4x4-1.png\" alt=\"\" class=\"wp-image-10716\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-4x4-1.png 1523w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-4x4-1-300x164.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-4x4-1-1024x560.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-4x4-1-768x420.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-24x24-4x4-1-535x293.png 535w\" sizes=\"(max-width: 1523px) 100vw, 1523px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Square of Order 25 &#8211; Bimagic Square<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">Below is a <strong>pandiagonal bimagic<\/strong> square of order 25. It is made with equal sums pandiagonal magic squares of order 5.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3014\" height=\"1103\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-25x25-bi.png\" alt=\"\" class=\"wp-image-10717\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-25x25-bi.png 3014w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-25x25-bi-300x110.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-25x25-bi-1024x375.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-25x25-bi-768x281.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-25x25-bi-535x196.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-25x25-bi-1536x562.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-25x25-bi-2048x749.png 2048w\" sizes=\"(max-width: 3014px) 100vw, 3014px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Square of Order 27<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 27=3^3. In this case we have pandiagonal magic square of order 27 with equal sums blocks of order 3. <\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1955\" height=\"1104\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-27x27-3x3-1.png\" alt=\"\" class=\"wp-image-10718\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-27x27-3x3-1.png 1955w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-27x27-3x3-1-300x169.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-27x27-3x3-1-1024x578.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-27x27-3x3-1-768x434.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-27x27-3x3-1-535x302.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-27x27-3x3-1-1536x867.png 1536w\" sizes=\"(max-width: 1955px) 100vw, 1955px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">Above we have considered blocks of order 9 formed by equal sums semi-magic squares of order 3. The blocks of order 9 are also equal sums and semi-magic squares.<\/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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 28<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 28=4&#215;7. This allow us to write pandiagonal magic squares of 28 with blocks of order 4 and 7. The first magic square is with equal sums pandiagonal magic squares of order 4. The second magic squares is with different sums pandiagonal magic squares of order 7.<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2196\" height=\"1172\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-4x4-2.png\" alt=\"\" class=\"wp-image-10722\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-4x4-2.png 2196w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-4x4-2-300x160.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-4x4-2-1024x547.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-4x4-2-768x410.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-4x4-2-535x286.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-4x4-2-1536x820.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-4x4-2-2048x1093.png 2048w\" sizes=\"(max-width: 2196px) 100vw, 2196px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2189\" height=\"1173\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-7x7-1.png\" alt=\"\" class=\"wp-image-10720\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-7x7-1.png 2189w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-7x7-1-300x161.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-7x7-1-1024x549.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-7x7-1-768x412.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-7x7-1-535x287.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-7x7-1-1536x823.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-28x28-7x7-1-2048x1097.png 2048w\" sizes=\"(max-width: 2189px) 100vw, 2189px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Square of Order 32<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">Below is a pandiagonal magic square of order 32, constructed with equal sums of pandiagonal magic squares of order 4.<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2756\" height=\"1331\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-32x32-4x4-1.png\" alt=\"\" class=\"wp-image-10727\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-32x32-4x4-1.png 2756w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-32x32-4x4-1-300x145.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-32x32-4x4-1-1024x495.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-32x32-4x4-1-768x371.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-32x32-4x4-1-535x258.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-32x32-4x4-1-1536x742.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-32x32-4x4-1-2048x989.png 2048w\" sizes=\"(max-width: 2756px) 100vw, 2756px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 33<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 33=3&#215;11. This allows us to write pandigonal magic squares with blocks of order 3 and 11. The first magic square is made with equal sums semi-magic squares of order 3. The second magic square is made with equal sums pandiagonal magic squares of order 11. See below:<\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2838\" height=\"1243\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-3x3-1.png\" alt=\"\" class=\"wp-image-10728\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-3x3-1.png 2838w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-3x3-1-300x131.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-3x3-1-1024x448.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-3x3-1-768x336.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-3x3-1-535x234.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-3x3-1-1536x673.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-3x3-1-2048x897.png 2048w\" sizes=\"(max-width: 2838px) 100vw, 2838px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2838\" height=\"1228\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-11x11-1.png\" alt=\"\" class=\"wp-image-10729\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-11x11-1.png 2838w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-11x11-1-300x130.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-11x11-1-1024x443.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-11x11-1-768x332.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-11x11-1-535x231.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-11x11-1-1536x665.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-33x33-11x11-1-2048x886.png 2048w\" sizes=\"(max-width: 2838px) 100vw, 2838px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 35<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 35=5&#215;7. This allows us to write pandigonal magic squares with blocks of orders 5 and 7. The first magic square of order 35 given below is made with equal sums pandiagonal magic squares of order 5. The second magic square of order 35 given below is made with equal sums pandiagonal magic squares of order 7. See below: <\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3002\" height=\"1298\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-5x5-1.png\" alt=\"\" class=\"wp-image-10730\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-5x5-1.png 3002w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-5x5-1-300x130.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-5x5-1-1024x443.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-5x5-1-768x332.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-5x5-1-535x231.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-5x5-1-1536x664.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-5x5-1-2048x886.png 2048w\" sizes=\"(max-width: 3002px) 100vw, 3002px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3000\" height=\"1300\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-7x7-1.png\" alt=\"\" class=\"wp-image-10731\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-7x7-1.png 3000w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-7x7-1-300x130.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-7x7-1-1024x444.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-7x7-1-768x333.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-7x7-1-535x232.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-7x7-1-1536x666.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-35x35-7x7-1-2048x887.png 2048w\" sizes=\"(max-width: 3000px) 100vw, 3000px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 36<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 36=3x3x4. This allows us to write pandigonal magic squares with blocks of orders 3 and 4. The first two magic square of order 36 given below are made with equal sums pandiagonal magic squares of order 12 and different sums magic squares of order 3. Each pandiagonal magic square of order 12 is formed by different sums magic squares of order 3. The third magic square of order 36 given below is made with equal sums pandiagonal magic squares of order 4. See below: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2929\" height=\"1334\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-3x3-1.png\" alt=\"\" class=\"wp-image-10734\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-3x3-1.png 2929w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-3x3-1-300x137.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-3x3-1-1024x466.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-3x3-1-768x350.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-3x3-1-535x244.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-3x3-1-1536x700.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-3x3-1-2048x933.png 2048w\" sizes=\"(max-width: 2929px) 100vw, 2929px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2932\" height=\"1332\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-12x12-3x3-1.png\" alt=\"\" class=\"wp-image-10735\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-12x12-3x3-1.png 2932w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-12x12-3x3-1-300x136.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-12x12-3x3-1-1024x465.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-12x12-3x3-1-768x349.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-12x12-3x3-1-535x243.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-12x12-3x3-1-1536x698.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-12x12-3x3-1-2048x930.png 2048w\" sizes=\"(max-width: 2932px) 100vw, 2932px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2932\" height=\"1333\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-4x4-1.png\" alt=\"\" class=\"wp-image-10736\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-4x4-1.png 2932w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-4x4-1-300x136.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-4x4-1-1024x466.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-4x4-1-768x349.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-4x4-1-535x243.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-4x4-1-1536x698.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-36x36-4x4-1-2048x931.png 2048w\" sizes=\"(max-width: 2932px) 100vw, 2932px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 39<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 39=3&#215;11. This allows us to write pandigonal magic squares with blocks of orders 3 and 13. The first magic square of order 39 given below is made with equal sums semi-magic squares of order 3.  The second magic square of order 39 is made with equal sums pandiagonal magic squares of order 13. See below: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3323\" height=\"1436\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-3x3-1.png\" alt=\"\" class=\"wp-image-10738\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-3x3-1.png 3323w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-3x3-1-300x130.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-3x3-1-1024x443.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-3x3-1-768x332.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-3x3-1-535x231.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-3x3-1-1536x664.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-3x3-1-2048x885.png 2048w\" sizes=\"(max-width: 3323px) 100vw, 3323px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3324\" height=\"1441\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-13x13-1.png\" alt=\"\" class=\"wp-image-10739\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-13x13-1.png 3324w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-13x13-1-300x130.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-13x13-1-1024x444.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-13x13-1-768x333.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-13x13-1-535x232.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-13x13-1-1536x666.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-39x39-13x13-1-2048x888.png 2048w\" sizes=\"(max-width: 3324px) 100vw, 3324px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 40<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 40=2x4x5. This allows us to write pandigonal magic squares with blocks of orders 4 and 5.   The first magic square of order 40 is made with equal sums pandiagonal magic squares of order 4. The second magic square of order 40 is made with 4 equal sums pandiagonal magic squares of order 20, where each pandiagonal magic square of order 20 is made with different sums magic squares of order 5. See below: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3400\" height=\"1468\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-4x4-1.png\" alt=\"\" class=\"wp-image-10740\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-4x4-1.png 3400w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-4x4-1-300x130.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-4x4-1-1024x442.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-4x4-1-768x332.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-4x4-1-535x231.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-4x4-1-1536x663.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-4x4-1-2048x884.png 2048w\" sizes=\"(max-width: 3400px) 100vw, 3400px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3069\" height=\"1349\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-5x5-2.png\" alt=\"\" class=\"wp-image-10742\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-5x5-2.png 3069w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-5x5-2-300x132.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-5x5-2-1024x450.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-5x5-2-768x338.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-5x5-2-535x235.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-5x5-2-1536x675.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-40x40-5x5-2-2048x900.png 2048w\" sizes=\"(max-width: 3069px) 100vw, 3069px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 44<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 44=4&#215;11. This allows us to write pandigonal magic squares with blocks of orders 4 and 11.   The first magic square of order 44 is made with equal sums pandiagonal magic squares of order 4. The second magic square of order 44 is made with equal sums pandiagonal magic squares of order 11. See below: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3461\" height=\"1757\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-44x44-4x4-1.png\" alt=\"\" class=\"wp-image-10743\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-44x44-4x4-1.png 3461w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-44x44-4x4-1-300x152.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-44x44-4x4-1-1024x520.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-44x44-4x4-1-768x390.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-44x44-4x4-1-535x272.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-44x44-4x4-1-1536x780.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-44x44-4x4-1-2048x1040.png 2048w\" sizes=\"(max-width: 3461px) 100vw, 3461px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3135\" height=\"1569\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-44x44-11x11-1.png\" alt=\"\" class=\"wp-image-10744\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-44x44-11x11-1.png 3135w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-44x44-11x11-1-300x150.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-44x44-11x11-1-1024x512.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-44x44-11x11-1-768x384.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-44x44-11x11-1-535x268.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-44x44-11x11-1-1536x769.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-44x44-11x11-1-2048x1025.png 2048w\" sizes=\"(max-width: 3135px) 100vw, 3135px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 45<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 45=3x3x5. This allows us to write pandigonal magic squares with blocks of orders 3 and 5.   The first magic square of order 45 is made with equal sums semi-magic squares of order 3. The second magic square is made with equal sums pandiagonal magic squares of order 15, where the each magic squares order 15 is made with equal sums pandiagonal magic squares of order 5. After 15, it is second magic square where both the blocks of order 3 and 5 are of equal sums. See below: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3431\" height=\"1503\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-3x3-1.png\" alt=\"\" class=\"wp-image-10745\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-3x3-1.png 3431w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-3x3-1-300x131.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-3x3-1-1024x449.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-3x3-1-768x336.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-3x3-1-535x234.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-3x3-1-1536x673.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-3x3-1-2048x897.png 2048w\" sizes=\"(max-width: 3431px) 100vw, 3431px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3437\" height=\"1510\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-15x15-5x5-1.png\" alt=\"\" class=\"wp-image-10746\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-15x15-5x5-1.png 3437w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-15x15-5x5-1-300x132.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-15x15-5x5-1-1024x450.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-15x15-5x5-1-768x337.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-15x15-5x5-1-535x235.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-15x15-5x5-1-1536x675.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-45x45-15x15-5x5-1-2048x900.png 2048w\" sizes=\"(max-width: 3437px) 100vw, 3437px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 48<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 48=3x4x4. This allows us to write pandigonal magic squares with blocks of orders 3 and 4.   The first magic square of order 48 is made with equal sums pandiagonal magic squares of order 12, where each pandiagonal magic square is made with different sums magic squares of order 3. The second magic square of order 48 is made with equal sums pandiagonal magic squares of order 4. See below:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3453\" height=\"1707\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-12x12-3x3-1.png\" alt=\"\" class=\"wp-image-10747\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-12x12-3x3-1.png 3453w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-12x12-3x3-1-300x148.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-12x12-3x3-1-1024x506.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-12x12-3x3-1-768x380.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-12x12-3x3-1-535x264.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-12x12-3x3-1-1536x759.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-12x12-3x3-1-2048x1012.png 2048w\" sizes=\"(max-width: 3453px) 100vw, 3453px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3449\" height=\"1701\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-4x4-1.png\" alt=\"\" class=\"wp-image-10748\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-4x4-1.png 3449w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-4x4-1-300x148.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-4x4-1-1024x505.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-4x4-1-768x379.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-4x4-1-535x264.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-4x4-1-1536x758.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/Pan-48x48-4x4-1-2048x1010.png 2048w\" sizes=\"(max-width: 3449px) 100vw, 3449px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Bimagic Square of Order 49<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 49=7^2. This allows us to write pandigonal magic squares with blocks of order 7. The magic square below is bimagic with equal sums pandiagonal magic squares of order 7. See below:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"3669\" height=\"1541\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-49x49-bi.png\" alt=\"\" class=\"wp-image-10758\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-49x49-bi.png 3669w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-49x49-bi-300x126.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-49x49-bi-1024x430.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-49x49-bi-768x323.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-49x49-bi-535x225.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-49x49-bi-1536x645.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-49x49-bi-2048x860.png 2048w\" sizes=\"(max-width: 3669px) 100vw, 3669px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 51<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 51=3&#215;17. This allows us to write pandigonal magic squares with blocks of orders 3 and 17. The first magic square of order 51 is made with equal sums semi-magic squares of order 3.  The second magic square of order 39 is made with equal sums pandiagonal magic squares of order 17. See below: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2491\" height=\"1538\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-3x3-1.png\" alt=\"\" class=\"wp-image-10759\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-3x3-1.png 2491w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-3x3-1-300x185.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-3x3-1-1024x632.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-3x3-1-768x474.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-3x3-1-535x330.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-3x3-1-1536x948.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-3x3-1-2048x1264.png 2048w\" sizes=\"(max-width: 2491px) 100vw, 2491px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2492\" height=\"1540\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-17x17-1.png\" alt=\"\" class=\"wp-image-10760\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-17x17-1.png 2492w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-17x17-1-300x185.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-17x17-1-1024x633.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-17x17-1-768x475.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-17x17-1-535x331.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-17x17-1-1536x949.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-51x51-17x17-1-2048x1266.png 2048w\" sizes=\"(max-width: 2492px) 100vw, 2492px\" \/><\/figure>\n<\/div>\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(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Pandiagonal Magic Squares of Order 52<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(122,220,180) 0%,rgb(255,255,255) 49%,rgb(0,208,130) 100%)\">We can write 52=4&#215;13. This allows us to write pandigonal magic squares with blocks of orders 4 and 13. The first magic square of order 52 is made with equal sums pandiagonal magic squares of order 4.  The second magic square of order 42 is made with unequal sums pandiagonal magic squares of order 13. See below: <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2541\" height=\"1565\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-4x4-1.png\" alt=\"\" class=\"wp-image-10761\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-4x4-1.png 2541w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-4x4-1-300x185.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-4x4-1-1024x631.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-4x4-1-768x473.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-4x4-1-535x330.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-4x4-1-1536x946.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-4x4-1-2048x1261.png 2048w\" sizes=\"(max-width: 2541px) 100vw, 2541px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2540\" height=\"1566\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-13x13-1.png\" alt=\"\" class=\"wp-image-10762\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-13x13-1.png 2540w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-13x13-1-300x185.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-13x13-1-1024x631.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-13x13-1-768x473.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-13x13-1-535x330.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-13x13-1-1536x947.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/pan-52x52-13x13-1-2048x1263.png 2048w\" sizes=\"(max-width: 2540px) 100vw, 2540px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(255,206,236) 0%,rgb(255,255,255) 59%,rgb(152,150,240) 100%)\">Proceeding in the similar way we can write pandiagonal magic squares of higher orders, except the multiple of 4k+2, k=1, 2, 3, &#8230;., i.e. of orders 6, 10, 14, etc.  Below is are excel files for down the whole work on pandiagonal magic squares of among the orders 4 to 108. The procedure used to write these pandiagonal magic squares is given in the reference list given in the beginning.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 51%,rgb(255,105,0) 100%)\">Excel file for Download of Pandiagonal Magic Squares among the Orders 4 to 108.<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-10541132-c048-4e38-9a17-e729c5d025ac\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/4to108-pan-online.xlsx\">4to108-pan-online<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/4to108-pan-online.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-10541132-c048-4e38-9a17-e729c5d025ac\">Download<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"has-text-align-justify has-background wp-block-paragraph\" style=\"background:linear-gradient(135deg,rgb(255,206,236) 0%,rgb(255,255,255) 59%,rgb(152,150,240) 100%)\">Below are some extra files of equal equal sums blocks of magic squares resulting in pandiagonal magic squares. These blocks of orders 3, 5 and 7.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Order 3&#215;3<\/strong><\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-bfe21cc1-bcb8-45af-8a14-6bdafd8b33c0\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/3x3-117-9-pandiagonal.xlsx\">3&#215;3-117-9-pandiagonal<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/3x3-117-9-pandiagonal.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-bfe21cc1-bcb8-45af-8a14-6bdafd8b33c0\">Download<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Order 5&#215;5<\/strong><\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-016468f7-7a59-4d8f-b1a0-59f450c02a42\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/5x5-145-15-pandiagonal.xlsx\">5&#215;5-145-15-pandiagonal<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/5x5-145-15-pandiagonal.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-016468f7-7a59-4d8f-b1a0-59f450c02a42\">Download<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Order 7&#215;7<\/strong><\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-d32907dc-d906-4d06-adc4-88f7a2ae14b5\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/7x7-133-21-pandiagonal.xlsx\">7&#215;7-133-21-pandiagonal<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/10\/7x7-133-21-pandiagonal.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-d32907dc-d906-4d06-adc4-88f7a2ae14b5\">Download<\/a><\/div>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This work brings pandiagonal magic squares in different situations. These are multiples of order 3, 4, 5, etc. We can always write pandiagonal magic squares multiples of any order, except of type 4k+2, where k=1, 2, 3, &#8230; etc. These orders are 6, 10, 14, 18, etc. Below are magic squares of orders 3, 4, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":10767,"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-10686","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\/10\/Pan-15x15-3x3-2.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10686","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=10686"}],"version-history":[{"count":13,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10686\/revisions"}],"predecessor-version":[{"id":10769,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/10686\/revisions\/10769"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/10767"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10686"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10686"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}