# Bimagic Squares of Orders 400, 800, 1600 and 2000: Blocks of Order 16

#### ByInder J. Taneja

Dec 23, 2023

Whole the work is done manually on excel sheets.

Below are bimagic squares written in blocks multiples of orders 16. This work brings bimagic squares multiples of 16, such as, of orders 400, 800, 1600 and 2000. Most of the work is done manually by author in 2011. See the reference 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.

Before presenting bimagic squares of orders 400, 800, 1600 and 2000, below is an example of bimagic square of order 16.

### Bimagic Square of Order 16

In this case, we have

S16×16 := 2056 and Sb16×16 := 351576,

See below the bimagic square of order 16.

Magic squares of order 4 are of equal sums, S4×4 :=(1/4) S16×16=514.

### Bimagic Square of Order 400: Blocks of Order 16

Bimagic square of order 400×400 has the the following properties:

1. 4×4 are magic squares with equal magic sums, i.e., S4×4:=320002
2. 16×16 are bimagic squares with equal magic and different bimagic sums, i.e., S16×16:=1280008.
3. 80×80 are bimagic squares with equal magic and different bimagic sums, i.e., S80×80:=6400040.
4. 400×400 is a bimagic square with magic and bimagic sums, i.e., S256×256:=32000200 and Sb256×256:=3413365333400.

Summarizing, we have a bimagic square of order 400×400, where blocks of orders 4×4, 16×16 and 80×80 are of equal magic sums. The magic squares of orders 16×16 and 80×80 are bimagic with different bimagic sums. These values are given in tables in an excel sheet attached with the work.

### Bimagic Square of Order 800: Blocks of Order 16

Bimagic square of order 800×800 has the the following properties:

1. 4×4 are magic squares with equal magic sums, i.e., S4×4:=1280002
2. 16×16 are bimagic squares with equal magic and different bimagic sums, i.e., S16×16:=5120008.
3. 160×160 are bimagic squares with equal magic and different bimagic sums, i.e., S160×160:=51200080.
4. 800×800 is a bimagic square with magic and bimagic sums, i.e., S800×800:=256000400 and Sb800×800:=109226922666800.

Summarizing, we have a bimagic square of order 800×800, where blocks of orders 4×4, 16×16 and 160×160 are of equal magic sums. The magic squares of orders 16×16 and 160×160 are bimagic with different bimagic sums. These values are given in tables in an excel sheet attached with the work.

### Bimagic Square of Order 1600: Blocks of Order 16

Bimagic square of order 1600×1600 has the the following properties:

1. 4×4 are magic squares with equal magic sums, i.e., S4×4:=5120002
2. 16×16 are bimagic squares with equal magic and different bimagic sums, i.e., S16×16:=20480008.
3. 80×80 are bimagic squares with equal magic and different bimagic sums, i.e., S80×80:=102400040.
4. 400×400 are bimagic squares with equal magic and different bimagic sums, i.e., S400×400:=512000200.
5. 1600×1600 is a bimagic square with magic and bimagic sums, i.e., S1600×1600:=2048000800 and Sb1600×1600:=3495255381333600.

Summarizing, we have a bimagic square of order 1600×1600, where blocks of orders 4×4, 16×16, 80×80 and 400×400 are of equal magic sums. The magic squares of orders 16×16, 80×80 and 400×400 are bimagic with different bimagic sums. These values are given in tables in an excel sheet attached with the work.

### Bimagic Square of Order 2000: Blocks of Order 16

Bimagic square of order 2000×2000 has the the following properties:

1. 4×4 are magic squares with equal magic sums, i.e., S4×4:=8000002
2. 16×16 are bimagic squares with equal magic and different bimagic sums, i.e., S16×16:=32000008.
3. 80×80 are bimagic squares with equal magic and different bimagic sums, i.e., S80×80:=160000040.
4. 400×400 are bimagic squares with equal magic and different bimagic sums, i.e., S400×400:=800000200.
5. 1600×1600 is a bimagic square with magic and bimagic sums, i.e., S2000×2000:=4000001000 and Sb2000×2000:=10666670666666800.

Summarizing, we have a bimagic square of order 2000×2000, where blocks of orders 4×4, 16×16, 80×80 and 400×400 are of equal magic sums. The magic squares of orders 16×16, 80×80 and 400×400 are bimagic with different bimagic sums. These values are given in tables in an excel sheet attached with the work.

### References:

1. Inder J. Taneja, Block-Wise Construction of Bimagic Squares: Multiples of Orders 8 and 16.
2. Inder J. Taneja, Block-Wise Construction of Bimagic Squares Multiples of 25: Orders 25, 125 and 625.
3. Inder J. Taneja, Block-Wise Construction of Bimagic Squares Multiples of 9: Orders 9, 81 and 729.
4. Inder J. Taneja, Block-Wise Construction of Bimagic Squares of Orders 121 and 1331.
5. Inder J. Taneja, Bimagic Squares of Orders 256, 512 and 1024: Blocks of Order 16.
6. Inder J. Taneja, Bimagic Squares of Orders 200 and 1000: Blocks of Order 8
7. Inder J. Taneja, Bimagic Squares of Orders 400, 800, 1600 and 2000: Blocks of Order 16. (This work)