{"id":9768,"date":"2023-07-17T21:30:34","date_gmt":"2023-07-18T00:30:34","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=9768"},"modified":"2023-07-25T02:10:25","modified_gmt":"2023-07-25T05:10:25","slug":"block-wise-bordered-magic-squares-multiples-of-15","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=9768","title":{"rendered":"Bordered Magic Squares Multiples of 15"},"content":{"rendered":"\n

Recently, the author revised and improved the block-wisebordered<\/strong> magic squares multiples of even number blocks<\/strong>. This means, multiples of 4, 6, 8, etc. The work is up to order 20. These works can be accessed at the following links.

1. Bordered and Pandiagonal Magic Squares Multiples of 4<\/a>.
2. Bordered Magic Squares Multiples of Magic Squares of Order 6<\/a>.
3. Bordered and Pandiagonal Magic Squares Multiples of 8 \u2013 Revised<\/a>
4. Bordered Magic Squares Multiples of 10 \u2013 Revised.<\/a>
5. Bordered and Pandiagonal Magic Squares Multiples of 12 \u2013 Revised.<\/a>
6. Bordered Magic Squares Multiples of 14 \u2013 Revised.<\/a>
7. Bordered and Pandiagonal Magic Squares Multiples of 16<\/a>.
8. Bordered Magic Squares Multiples of 18<\/a>.
9. Bordered and Pandiagonal Magic Squares Multiples of 20.<\/a>

Author also worked with odd number blocks. The work is for multiples of order 3, 5, 7, 9, 11 and 13. See the links below:
1. Bordered and Pandiagonal Magic Squares Multiples of 3<\/a>.
2. Bordered and Pandiagonal Magic Squares Multiples of 5.<\/a>
3. Bordered and Pandiagonal Magic Squares Multiples of 7<\/a>.
4. Bordered Magic Squares Multiples of 9 \u2013 Revised<\/a>.
5. Bordered Magic Squares Multiples of Order 11- Revised<\/a>.
6. Bordered Magic Squares Multiples of 13<\/a>.

The advantage in studying block-wise bordered<\/strong> magic squares is that when we remove external borders, still we left with magic squares with sequential entries. <\/p>\n\n\n\n

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We observe that the above study of even order blocks starting from 4. Recently, author studied the double digit<\/strong> or two digits<\/strong> borders resulting in interesting magic squares. These studies can be accessed at the following link:
1. Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24<\/a>.
2. Two Digits Bordered Magic Squares of Orders 28 and 32<\/a>.
3. Two digits Bordered Magic Squares of Order 36.<\/a>
4. Two digits Bordered Magic Squares of Order 40.<\/a>

The work is for the order of type 4k+2, where k>1, i.e., for the orders 10, 14, 18, 22, 26 and 30 can be access at the following links:
1. Two digits Bordered Magic Squares of Orders 10, 14, 18 and 22<\/a>.
2. Two digits Bordered Magic Squares o<\/a>f Orders 26 and 30<\/a>.
3. Two Digits Bordered Magic Squares of Orders 28 and 32.<\/a>

Some studies towards cornered magic squares are also made. See the following links:
1. Cornered Magic Squares of Orders 5 to 10<\/a>.
2. Cornered Magic Squares of Orders 11 to 13<\/a>.
3. Cornered Magic Squares of Orders 14 to 24<\/a>.

Below are examples of bordered magic squares multiples of order 15. These examples are only for the magic squares of orders 15, 45 and 60. Total work is with 36 different types of magic squares of order 15. PDF files<\/strong> of magic squares of order 15, 45 and 60 are also attached. In the end there are excel files of whole the up to order 150 multiples of 15. <\/p>\n\n\n\n

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The work can also be accessed at the following link:
Inder J. Taneja<\/strong>, Bordered Magic Squares Multiples of 15, Zenodo<\/strong>, July 24, pp. 1-35, 2023, https:\/\/doi.org\/10.5281\/zenodo.8178935<\/a>.<\/p>\n\n\n\n

Magic Squares of Order 15<\/h4>\n\n\n\n

Below are 36 magic square of order 15. We have used these magic squares to bring bordered magic squares up to order 150 as multiples of 15.<\/p>\n\n\n\n

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Pdf file of Magic Squares of Order 15<\/h3>\n\n\n\n

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