Panmagic 4x4 inlay square inside a 6x6 magic square

 

An inlay square is a (not pure) magic square inside a larger (pure) magic square.

 

The smallest even inlay square is a (not pure) 4x4 square inside a (pure) 6x6 square. In a pure 6x6 square are the digits from 1 up to 36.

 

 

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

 

 

27

28

29

30

31

32

33

34

35

36

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

 

 

To construct the 4x4 inlay square, use the middle 16 digits (= yellow marked). Take for example the first basic panmagic 4x4 square and add 10 to each digit.

 

 

 Pure 4x4 +10           =   (not pure) inlay 4x4

1

8

13

12

   

11

18

23

22

15

10

3

6

   

25

20

13

16

4

5

16

9

   

14

15

26

19

14

11

2

7

   

24

21

12

17

 

  

There are the following three boundary conditions to construct the border of the pure 6x6 magic square:

  

A positive digit in each row/column/diagonal must be put opposite the same negative digit (= translation of the digits 27 up to 36);

  

In the top row, the bottom row, the left column and the right column must be put 3x positive and 3x negative digit;

  

● The digits of the top row, the bottom row, the left column and the right column must each time sum to 0.

  

Make the top row and the right column and start top left (for example) with the digit 1. Because the digit top left is 1, the digit -1 must be put right down. The digit to be filled in top right (?) is as well in the top row as the right column.

 

 

1

       

?

           
           
           
           
         

-1

 

 

The sum of 1 up to 10 is 55. You need the digits 1 up to 10 plus two digits double. Take the digits 1 and 8 double, than the sum of each 3 digits must be: [55 + 1 + 8] / 4 = 16.

 

1+5+10

1+6+ 9

1+7+ 8

2+4+10

2+5+ 9

2+6+ 8

3+4+ 9

3+5+ 8

3+6+ 7

4+5+ 7

 

 

Above I puzzled 4x 3 possibilities. The digits can be filled in as follows.

 

 

 Fill in 4 possibilities                      Fill in opposite digits

1

6

9

-3

-5

-8

   

1

6

9

-3

-5

-8

         

2

   

-2

       

2

         

4

   

-4

       

4

         

10

   

-10

       

10

         

-7

   

7

       

-7

         

-1

   

8

-6

-9

3

5

-1

 

 
The result is:

  

 

1

6

9

34

32

29

35

11

18

23

22

2

33

25

20

13

16

4

27

14

15

26

19

10

7

24

21

12

17

30

8

31

28

3

5

36

 

 

You can use this method to construct larger (singular) even inlay squares (e.g. a 6x6 inlay inside an 8x8 square, 8x8 inlay inside a 10x10 square, 10x10 inlay inside a 12x12 square, …).

 

 

Download
6x6, Panmagic 4x4 in 6x6.xls
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