{"id":14561,"date":"2025-01-15T19:03:08","date_gmt":"2025-01-15T22:03:08","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=14561"},"modified":"2025-01-23T01:56:58","modified_gmt":"2025-01-23T04:56:58","slug":"water-reflection-magic-squares-order-20","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=14561","title":{"rendered":"Water Reflection Magic Squares: Orders 17 to 20"},"content":{"rendered":"\n

This work brings more concepts in magic squares. In past we studied a lot of upside-down and mirror looking magic squares. These are based some kind of digital\/special fonts. To understand better let’s consider the following 10 digits from 0 to 9:<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

Upside-down (180 degrees rotation)<\/strong><\/h4>\n\n\n\n

Let’s make 180 degrees rotation over the above 10 digits, we get<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

We observe that the numbers 0, 1, 2, 5, 6, 8 and 9 are still there. The difference is that 6 becomes 9.<\/p>\n\n\n\n

Mirror Looking (Horizontal Flip)<\/h4>\n\n\n\n

Let’s see how these numbers can seen in mirror:<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

We observe that the numbers 0, 1, 2, 5 and 8 are still there. In this case the numbers 2 and 5 interchanges, i.e., 2 becomes 5 and 5 as 2.<\/p>\n\n\n\n

In general there are two kinds of flips, i.e., horizontal flip<\/strong> and vertical filp<\/strong>. See the image below:<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

Source: https:\/\/www.mathsisfun.com\/definitions\/vertical-flip.html<\/a><\/p>\n\n\n\n

Water Reflection (Vertical Flip)<\/h4>\n\n\n\n

Making vertical flip over the digits 0 to 9, we get<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

We observe the numbers 0, 1, 2, 3, 5 and 8 remains the same. The numebrs 0, 1, 2, 5 and 8 are already in mirror looking except the number. That’s why we call these numbers as universal<\/strong> numbers. Here also 2 becomes 5 and 5 as 2. Thus using vertical flip<\/strong>, we get an extra number as 3. Let’s call the operation as water reflection<\/strong>. This means that the numbers 0, 1, 2, 3, 5 and 8 are water reflexive<\/strong>.<\/p>\n\n\n\n

The aim this work is write magic squares based on the numbers 0, 1, 2, 3, 5 and 8, where 3 is always there, i.e., making combinations of 3 with the numbers 0, 1, 2, 5 and 8. This work is for orders 14 to 16. Full work can be downloaded from the following link:<\/p>\n\n\n\n