Each magic sum

 

A pure magic (4x4) square with all the digits from 1 up to n x n (1 up to 16) has one fixed magic sum ([1+n x n]/2 x n), for example ([1 + 4 x 4]/2 x 4) = 34. You can skip the boundary of the pure magic square. In a not pure magic square all digits must be different, but it don't need to be consecutive digits. If the magic square is not pure it is possible to choose a magic sum. If you choose the right digits to fill in the magic square you get the magic sum you want. See below an impure 4x4 magic square with the magic sum of 0.

 

 

22

14

-46

10

-30

-6

6

30

26

2

-2

-26

-18

-10

42

-14

 

  

To get the magic sum of 403, divide 403 by 4 and you get 100 remainder 3. 

Add 100 to each cell and add the remainder of 3 extra to each yellow marked cell.

  

 

125

114

54

110

70

94

109

130

126

102

98

77

82

93

142

86

 

 

This square is not panmagic, but has extra magic features (for example 2x2 sub-squares, horizontally swinging and vertical "ears"). Discover it yourself.

 

See below the mathematical alternative.
  
 

 

a

c+3

d+1

b+2

d+2

b+1

a+3

c

b+3

d

c+2

a+1

c+1

a+2

b

d+3

 


a + b + c + d = magic sum -/- 6

 

Boundaries are: b - a >= 4 and c - b >= 4 and d - c >= 4 (to get always 16 different digits in the magic square)

 

Choose the magic sum, for example 100. Take care that a + b + c + d = 100 - 6 = 94 (and a, b, c and d must differ minimum 4), for example:
 
 

 

a =

7

b =

20

c =

31

d =

36

 

 

7

34

37

22

38

21

10

31

23

36

33

8

32

9

20

39

 
 
Notify that this square is "Dürer" magic (it has the same magic features as the Dürer magic square).

  

 

The idea of Aridwana from Indonesia is to convert names into numbers, using the table below.

 

 

A

1

B

2

C

3

D

4

E

5

F

6

G

7

H

8

I

9

J

10

K

20

L

30

M

40

N

50

O

60

P

70

Q

80

R

90

S

100

T

200

U

300

V

400

W

500

X

600

Y

700

Z

800

 

 

For example I have converted the names of myself, my wife and my two daughters:

 

 

Arie = A + R + I + E = 1 + 90 + 9 + 5 = 105
Conny = C + O + N + N + Y = 3 + 60 + 50 + 50 + 700 = 863
Iris = I + R + I + S = 9 + 90 + 9 + 100 = 208
Tamar = T + A + M + A + R = 200 + 1 + 40 + 1 + 90 = 332
 
 
                    4 x 4 family Breedijk magic square

   

Arie

335

864

210

 
   

865

209

108

Tamar

 
   

211

Conny

334

106

 
   

333

107

Iris

866

 

 

           
             
             
   

1514

1514

1514

1514

 
 

1514

       

1514

1514

 

105

335

864

210

 

1514

 

865

209

108

332

 

1514

 

211

863

334

106

 

1514

 

333

107

208

866

 



The magic sum of this Dürer magic square is 1514, the year in which Albrecht Dürer published his famous magic square!!!

 

 

Download
4x4, Each magic sum.xls
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4x4, Names & numbers magic square.xls
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