This work brings different ways representing magic squares in prime numbers. Initial aim is represent in terms of blocks. Some of possibilities led us to concentric or single-layer bordered magic squares. It includes blocks of orders 3, 4, 5, etc. Sometimes these blocks are of equal or sometimes of different magic sums. Initially this work is for the orders 6 to 29. In case of order 7, 11, 13, etc, we don’t have division in blocks but except 7, the magic squares type 11, 13, 17, 19 etc. are block-bordered. This means there are one or two layers are single-border with internal part as borders.
Whenever, we write prime magic square, it means that the magic squares are composed of prime distinct number entries except only in few cases, there are repetition of numbers but this repepetition is cleary explained.
This work is specially for distinct prime numbers, except in some cases there are repetition of internal values. For similar kind of work for sequential numbers entries, see the author’s work given in the reference list.
This work is also available online at the following link:
Prime Magic Squares of Order 6
Below are few examples of prime magic squares of order 9 with different representations.
Example 1.
It is a general prime magic square of order 6 without any division.
Example 2.
It is single-layer or cocentrc prime magic square. The internal block is again a magic square of order 4. See below:
Example 3.
It is block-wise prime magic square composed of four equal sums prime magic square of order 3. The central prime number in each case 1000003 is repeated, because it helps in bringing four equal sums prime magic square of order 3. See below:
Prime Magic Squares of Order 7
In this case, we have an example of cocentric prime magic square of order 7. See below
Example 1.
It is single-layer or cocentric prime magic square of order 7. The internal blocks are prime magic squares of orders 5 and 3. See below:
Prime Magic Squares of Order 8
Below are few examples of prime magic squares of order 8 with different representations.
Example 1.
It a normal prime magic square of order 8 without any special property
Example 2.
It is single-layer or cocentrc prime magic square of order 8. The internal blocks are prime magic squaers of orders 6 and 4. See below:
Example 3.
It is block-wise prime magic square of order 8 with four equal sums prime magic squares of order 4. See below
Prime Magic Squares of Order 9
Below are few examples of prime magic squares of order 9 with different representations.
Example 1.
It is general prime magic square of order 9 without any special representation.
Example 2.
It is a block-wise prime magic square of order 9 with 9 blocks of different sums magic square of order 3. See below few examples of order 3:
Example 4.
It is a block-wise prime magic square of order 9 with 9 blocks of equal sums magic square of order 3. It has a repreated entry 100003 in each block of order 3. See below few examples of order 3:
Example 4.
It is a single-layer or cocentric prime magic square of order 9. The internal blocks are prime magic squares of orders 7, 5 and 3. See below:
Prime Magic Squares of Order 10
Below are few examples of prime magic squares of order 10 with different representations.
Example 1.
It is a normal prime magic square of order 10 without any special representation.
Example 2.
It is a block-bordered prime magic square of order 10. The internal part is a magic square of order 8 composed of 4 equal sums prime magic squares of order 4. It is a same Example 3 as given before on prime magic square of order 8.
Example 3.
It is a single-layer or cocentric prime magic square of order 10. The internal blocks are prime magic squares of orders 8, 6 and 4. These are the same as given in Example 2 in prime magic squares of order 8.
Prime Magic Squares of Order 11
Below are few examples of prime magic squares of order 10 with different representations.
Example 1.
It is a block-bordered prime magic square with inner part as a prime magic square of order 9. It is composed of 9 different sums magic squares of order 3. See below
Example 2.
It is a single-layer or cocentric prime magic square of order 11. The internal blocks are prime magic squares of orders 9, 7, 5 and 3. See below
Prime Magic Squares of Order 12
Below are few examples of prime magic squares of order 12 with different representations.
Example 1.
It is a block-wise prime magic square of order 12 with 16 blocks of different sums prime magic square of order 3. See below few examples of order 3:
Example 2.
It is a block-wise prime magic square of order 12 with 16 blocks of equal sums magic square of order 3. It has a repreated entry 100003 in each block of order 3. See below few examples of order 3:
Example 3.
It is a block-wise prime magic square of order 12 with 12 blocks of different sums prime magic square of order 4. See below few examples of order 4:
Example 4.
It is a block-wise prime magic square of order 12 with 16 blocks of equal sums magic square of order 4. See below few examples of order 4:
Example 5.
It is a block-wise prime magic square of order 12 with 4 blocks of equal sums magic square of order 6. These magic squares are cocentric typ. See below few examples of orders 6 and 4:
Example 6.
It is a single-layer or cocentric prime magic square of order 12. The internal blocks are prime magic squares of orders 10, 8, 6 and 4. The internal block of order 10 is the same as given section of prime magic squares of order 10.
Prime Magic Squares of Order 13
Below are few examples of prime magic squares of order 13 with different representations.
Example 1.
It is a block bordered prime magic square of order 13, where the interal block of 9 with different blocks of primes magic squares of order 3 is the as as given in section on prime magic squares of order 9. Moreover, the internal prime magic squre of order 11 is the same as given in section on prime magic squares of order 11.
Example 2.
It is a cocentric single-layer prime magic square of order 13. The internal cocentric single-layer prime magic squre of order 11 is the same as given in section on prime magic squares of order 11.
Prime Magic Squares of Order 14
Below are few examples of prime magic squares of order 14 with different representations.
Example 1
It is block-bordered prime magic square of order 14. The internal block is a prime magic square of order 12 composed different prime magic squares of order 3. It is the same as given in section on primes magic squares of order 12.
Example 2
It is block-bordered prime magic square of order 14. The internal block is a prime magic square of order 12 composed equal sums prime magic squares of order 3. It is the same as given in section on primes magic squares of order 12.
Example 3
It is block-bordered prime magic square of order 14. The internal block is a prime magic square of order 12 composed different prime magic squares of order 4. It is the same as given in section on primes magic squares of order 12.
Example 4
It is block-bordered prime magic square of order 14. The internal block is a prime magic square of order 12 composed equal sums prime magic squares of order 3. It is the same as given in section on primes magic squares of order 12.
Example 5
It is block-bordered prime magic square of order 14. The internal block is a prime magic square of order 12 composed different prime magic squares of order 6. It is the same as given in section on primes magic squares of order 12.
Example 6
It is block-wise prime magic square of order 14. It is composed of four equal sums single-layer bordered equal sums prime magic squares of order 7. The center point of each blocks of order 7 is same in all the four blocks, i.e., 1000003. See below two examples for verification:
Example 7
It is concentric single-layer bordered prime magic square of order 14. The internal concentric single-layer bordered prime magic square of order 12 is the same as given in section on prime magic squares of order 12.
References
- H. White, Magic Squares of Prime Numbers, https://budshaw.ca/PrimeMagicSquares.html
- Heinz, Harvey, Prime Numbers Magic Squares, http://recmath.org/Magic%20Squares/primesqr.htm
- Makarova, Natalia, Concentric magic squares of primes
http://primesmagicgames.altervista.org/wp/forums/topic/concentric-magic-squares-of-primes/ - Roberto C. Angelone, A Fully Nested 729 x 729 Unique-Prime Magic Square Constructed from Nine Correlated 243 x 243 Prime Magic Blocks, https://zenodo.org/records/20098521
- Inder J. Taneja, Single-Layer Bordered Prime Magic Squares of Even Orders – Recreating Numbers and Magic Squares
- Inder J. Taneja, Single-Layer Bordered Prime Magic Square of Odd Orders – Recreating Numbers and Magic Squares
- Inder J. Taneja, Block-Structured Prime Magic Square of Order 1220×1220 – Recreating Numbers and Magic Squares
- Inder J. Taneja, Block-Structured Prime Magic Squares of Orders 6 to 14 – Recreating Numbers and Magic Squares