Block-Wise Construction of Bimagic Squares Multiples of 9: Orders 9, 81 and 729

Below are bimagic squares written in blocks multiples of order 9. These are of orders 9, 81 and 729 The work is done manually by author in 2011. The work is summarized in the link below:

Inder J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011, pp. 1-14, (22.02.2011), https://doi.org/10.48550/arXiv.1102.3052.

Before proceeding further below are the basic formulas to check the sums of magic and bimagic squares

• Magic Sum

• Bimagic Sum

where n is the order the magic square.

Bimagic Square of Order 9

In this case, we have

S_9×9:=369; Sb_9×9:= 20049,

Sum of entries of blocks of order 3×3 are the same as of magic square. i.e., S_9×9:=369. Below is an example of bimagic square of order 9. It is constructed by G. Pfeffermann in 1891.

We shall use the idea of this bimagic square to bring bimagic squares of orders 81 and 729.

Bimagic Square of Order 81: Blocks of Order 9

Bimagic Square of order 81×81 with magic and bimagic sums

S_81×81:=265761 and Sb_81×81:=1162527201

Each block of order 9 is a magic square with equal magic sums, S_9×9:=29529. It also has the same property as of order 9, i.e., sum of entries of blocks of order 3×3 are the same as of magic square of order 9, i.e., S_9×9:=29529.

Excel sheet of Bimagic Square of Order 81

Bimagic Square of Order 729: Blocks of Order 9 and 81

Bimagic Square of order 729 with magic and bimagic sums

S_729×729 :=193710609 and Sb_729×729 := 68630571075249

Blocks of orders 9 and 81 are only magic squares with equal magic sums:

S_81×81 := 21523401 and S_9×9 := 2391489.

Similar to magic squares of orders 9 and 81 as given above, in this case also the sum of entries of blocks of order 3×3 are the same as of magic square of orderr 9, i.e., S_9×9 := 2391489.

Unfortunately, in this case, we don’t have sub-groups of orders 9 or 81 as bimagic squares. These are only magic squares of equal sums.

Excel sheet of Bimagic Square of Order 729

References

1. Inder J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011, pp. 1-14, (22.02.2011), https://doi.org/10.48550/arXiv.1102.3052.
2. Inder J. Taneja, Block-Wise Construction of Bimagic Squares: Multiples of Orders 8 and 16.
3. Inder J. Taneja, Block-Wise Construction of Bimagic Squares Multiples of 25: Orders 25, 125 and 625.
4. Inder J. Taneja, Block-Wise Construction of Bimagic Squares of Orders 49 and 343.
5. Inder J. Taneja, Block-Wise Construction of Bimagic Squares Multiples of 9: Orders 9, 81 and 729.
6. Inder J. Taneja, Block-Wise Construction of Bimagic Squares of Orders 121 and 1331.
7. Inder J. Taneja, Bimagic Squares of Orders 256, 512 and 1024: Blocks of Order 16.
8. Inder J. Taneja, Bimagic Squares of Orders 200 and 1000: Blocks of Order 8.
9. Inder J. Taneja, Bimagic Squares of Orders 400, 800, 1600 and 2000: Blocks of Order 16.
10. Inder J. Taneja, Bimagic Squares of Orders 100, 110 and 121: Blocks of Orders 10 and 11.
11. Inder J. Taneja, Universal and Upside-Down Magic and Bimagic Squares of Order 16
12. Inder J. Taneja, Universal and Upside-Down Magic and Bimagic Squares of Order 25