The 3 basic 4x4 panmagic squares

 

48 of the 880 pure magic 4x4 squares are panmagic (= group 1). These squares have (just like the larger most perfect magic squares) the following structure:

 

1

8

10

15

12

13

3

6

7

2

16

9

14

11

5

4

 


The sum of two digits of the same colour is each time (the lowest plus the highest digit of the magic square: 1+16=) 17.

 

You need to know only the following panmagic 4x4 basic squares, to construct all 48 panmagic 4x4 squares (excluding rotation and/or mirroring):

  

 

1

8

13

12

   

1

8

11

14

   

1

8

10

15

15

10

3

6

   

15

10

5

4

   

14

11

5

4

4

5

16

9

   

6

3

16

9

   

7

2

16

9

14

11

2

7

   

12

13

2

7

   

12

13

3

6

  

 

Make a 2x2 carpet of one of the basic squares and you can get all 16 squares by shifting on the carpet. See for example:

  

 

1

8

10

15

1

8

10

15

12

13

3

6

12

13

3

6

7

2

16

9

7

2

16

9

14

11

5

4

14

11

5

4

1

8

10

15

1

8

10

15

12

13

3

6

12

13

3

6

7

2

16

9

7

2

16

9

14

11

5

4

14

11

5

4

 

 

You can get 8 in stead of 1 solution of the above yellow marked square by rotating and/or mirroring:

  

 

yellow marked square

4

14

11

5

 

Mirroring

 

5

11

14

4

         

15

1

8

10

     

10

8

1

15

         

6

12

13

3

     

3

13

12

6

         

9

7

2

16

     

16

2

7

9

                               
                               

rotation by 1 quarter

9

6

15

4

 

Mirroring

 

4

15

6

9

         

7

12

1

14

     

14

1

12

7

         

2

13

8

11

     

11

8

13

2

         

16

3

10

5

     

5

10

3

16

                               
                               

rotation by 2 quarters

16

2

7

9

 

Mirroring

 

9

7

2

16

         

3

13

12

6

     

6

12

13

3

         

10

8

1

15

     

15

1

8

10

         

5

11

14

4

     

4

14

11

5

                               
                               

rotation by 3 quarters

5

10

3

16

 

Mirroring

 

16

3

10

5

         

11

8

13

2

     

2

13

8

11

         

14

1

12

7

     

7

12

1

14

         

4

15

6

9

     

9

6

15

4

 

 

There are 3 basic 4x4 panmagic squares. There are 16 possibilities by shifting on the carpet. There are 8 possibilities by rotating and/or mirroring. This gives in total 3 x 16 x 8 is 384 possibilities (including rotating and/or mirroring).

 

 

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4x4, the 3 basic panmagic 4x4 squares.xl
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