By boarded magic squares we understand that if we remove the external border still we are left with magic squares of lower orders with sequential entries. Single digits bordered magic squares are well-known. This work is for double digits not non in the literature. This work is already apresented by author in another work
- Inder J. Taneja, New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108.
This above link is of only of one type, i.e. general double digits bordered magic square. Below are two more types are given. Both of these are known by striped magic squres. One is opposite-type and another is circular type.
Order 7
The first magic square is general double digits bordered magic square. The second one is semi-striped magic squares, as it is constructed with equal width striped magic rectangles except the inner block of order 3. It is magic square of order 3. The third magic square is also semi-striped magic square as it is contructed with four equal sum equal width magic rectangles. It is semi-striped as the inner block os magic square of order 3. Summarizing, there are thre magic squares, the first one is general double digits bordered magic square. The other two are semi-stripled magic squares. The inner block is a magic squares of order 3.
Order 8
The first magic square is general double digits bordered magic square. The second one is striped magic square, as it is constructed with equal width striped magic rectangles. The third magic square is also striped magic square as it is also contructed with equal width magic rectangles. Summarizing the first magic square is general double digits bordered magic square. The last two are striped double digits bordered magic square
Order 9
The first magic square is general double digits bordered magic square. The second one is semi-striped magic squares, as it is constructed with equal width striped magic rectangles except the inner block of order 5. It is magic square of order 5. The third magic square is also semi-striped magic square as it is contructed with four equal sum equal width magic rectangles. It is also semi-striped as the inner block is magic square of order 5. Summarizing, we have one magic square as general double digits bordered magic square. The other two as semi-striped magic squares.
Order 10
The first magic square is general double digits bordered magic square. The second one is striped magic square, as it is constructed with equal width striped magic rectangles. The third magic square is also striped magic square as it is also contructed with four equal sum equal width magic rectangles. Summarizing the first magic square is general double digits bordered magic square. The last two are striped double digits bordered magic square.
Order 11
The first magic square is general double digits bordered magic square. The second one is semi-striped magic squares, as it is constructed with equal width striped magic rectangles except the inner block of order 3. It is magic square of order 3. The third magic square is also semi-striped magic square as it is contructed with four equal sum equal width magic rectangles. It is semi-striped as the inner block os magic square of order 3. Summarizing, there are thre magic squares, the first one is general double digits bordered magic square. The other two are semi-stripled magic squares. The inner block is a magic squares of order 3.
Order 12
The first magic square is general double digits bordered magic square. The second one is striped magic square, as it is constructed with equal width striped magic rectangles. The third magic square is also striped magic square as it is also contructed with equal width magic rectangles. Summarizing the first magic square is general double digits bordered magic square. The last two are striped double digits bordered magic square
Order 13
The first magic square is general double digits bordered magic square. The second one is semi-striped magic squares, as it is constructed with equal width striped magic rectangles except the inner block of order 5. It is magic square of order 5. The third magic square is also semi-striped magic square as it is contructed with four equal sum equal width magic rectangles. It is also semi-striped as the inner block is magic square of order 5. Summarizing, we have one magic square as general double digits bordered magic square. The other two as semi-striped magic squares.
Order 14
The first magic square is general double digits bordered magic square. The second one is striped magic square, as it is constructed with equal width striped magic rectangles. The third magic square is also striped magic square as it is also contructed with four equal sum equal width magic rectangles. Summarizing the first magic square is general double digits bordered magic square. The last two are striped double digits bordered magic square.
Order 15
The first magic square is general double digits bordered magic square. The second one is semi-striped magic squares, as it is constructed with equal width striped magic rectangles except the inner block of order 3. It is magic square of order 3. The third magic square is also semi-striped magic square as it is contructed with four equal sum equal width magic rectangles. It is semi-striped as the inner block os magic square of order 3. Summarizing, there are thre magic squares, the first one is general double digits bordered magic square. The other two are semi-stripled magic squares. The inner block is a magic squares of order 3.
Order 16
The first magic square is general double digits bordered magic square. The second one is striped magic square, as it is constructed with equal width striped magic rectangles. The third magic square is also striped magic square as it is also contructed with equal width magic rectangles. Summarizing the first magic square is general double digits bordered magic square. The last two are striped double digits bordered magic square
Order 17
The first magic square is general double digits bordered magic square. The second one is semi-striped magic squares, as it is constructed with equal width striped magic rectangles except the inner block of order 5. It is magic square of order 5. The third magic square is also semi-striped magic square as it is contructed with four equal sum equal width magic rectangles. It is also semi-striped as the inner block is magic square of order 5. Summarizing, we have one magic square as general double digits bordered magic square. The other two as semi-striped magic squares.
Order 18
We observe that in case of even order magic squares, the last two magic squares are stripled magic squares. In case of of odd order magic squares the last two types of magic squares are semi-striped magic squares, since there are magic squares of orders 3 or 5 as inner blocks of magic squres.
Moreover these are divided in four groups of orders as follows:
1. 7, 11, 15,…
2. 8, 12, 16,…
3. 9, 13, 17,…
4. 10, 14, 18,…
The total work is up to order 50. More examples of similar kind are given below in excel files.
Excel file for Download
References
- Inder J. Taneja, Two Digits Bordered Magic Squares Multiples of 4: Orders 8 to 24, Zenodo, April, 26, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.7866956.
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 28 and 32, Zenodo, April, 26, 2023, pp. 1-36, https://doi.org/10.5281/zenodo.7866981.
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 10, 14, 18 and 22, Zenodo, April, 30, 2023, pp. 1-43, https://doi.org/10.5281/zenodo.7880931.
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 26 and 30, Zenodo, April, 30, 2023, pp. 1-45, https://doi.org/10.5281/zenodo.7880937.
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 36 and 40, Zenodo, May, 04, 2023, pp. 1-41, https://doi.org/10.5281/zenodo.7896709.
- Inder J. Taneja, Two Digits Bordered Magic Squares of Orders 34 and 38, Zenodo, May 10, 2023, pp. 1-45, https://doi.org/10.5281/zenodo.7922571.
- Inder J. Taneja, New Concepts in Magic Squares: Double Digits Bordered Magic Squares of Orders 7 to 108, Zenodo, August 09, 2023, pp. 1-30, https://doi.org/10.5281/zenodo.8230214.