Magic Square – Equal sums and square sums

 
 

\begin{pmatrix}   \;a \; & \; b \; & \; c \; \\   \;d \; & \; e \; & \; f \; \\   \;g \; & \; h \; & \; k \;  \end{pmatrix}

det \; = \; a \, e \, k \; - \; a \, f \, h \; - \; b \, d \, k \; + \; b \, f \, g \; + \; c \, d \, h \; - \; c \, e \, g

 

\begin{pmatrix}   \;4 \; & \; 3 \; & \; 8 \; \\   \;9 \; & \; 5 \; & \; 1 \; \\   \;2 \; & \; 7 \; & \; 6 \;  \end{pmatrix}           det = 360

 

4^2 + 3^2 + 8^2 = 89   ……….   4^2 + 9^2 + 2^2 = 101
9^2 + 5^2 + 1^2 = 107   ………   3^2 + 5^2 + 7^2 = 83
2^2 + 7^2 + 6^2 = 89   ……….   8^2 + 1^2 + 6^2 = 101

4^2 + 5^2 + 6^2 = 77   ……….   8^2 + 5^2 + 2^2 = 93
2^2 + 3^2 + 1^2 = 14   ……….   1^2 + 7^2 + 4^2 = 66
9^2 + 7^2 + 8^2 = 194   ………   6^2 + 3^2 + 9^2 = 126

(4\times 9) + (3\times 5) + (8\times 1) = \; 59 \; = (9\times 2) + (5\times 7) + (1\times 6)
(4\times 3) + (9\times 5) + (2\times 7) = \; 71 \; = (3\times 8) + (5\times 1) + (7\times 6)
(4\times 5) + (9\times 7) + (2\times 3) = \; 89 \; = (5\times 6) + (7\times 8) + (3\times 1)
(8\times 5) + (1\times 7) + (6\times 3) = \; 65 \; = (5\times 2) + (7\times 4) + (3\times 9)
 

Rows:

(4 \, x^2 + 3 \, x + 8) \; + \; (9 \, x^2 + 5 \, x + 1) \; + \; (2 \, x^2 + 7 \, x + 6)
= \; (8 \, x^2 + 3 \, x + 4) \; + \; (x^2 + 5 \, x + 9) \; + \; (6 \, x^2 + 7 \, x + 2)
= \; 15 \, x^2 \; + \; 15 \, x \; + \; 15

(4 \, x^2 + 3 \, x + 8)^2 \; + \; (9 \, x^2 + 5 \, x + 1)^2 \; + \; (2 \, x^2 + 7 \, x + 6)^2
= \; (8 \, x^2 + 3 \, x + 4)^2 \; + \; (x^2 + 5 \, x + 9)^2 \; + \; (6 \, x^2 + 7 \, x + 2)^2
= \; 101 \, x^4 \; + \; 142 \, x^3 \; + \; 189 \, x^2 \; + \; 142 \, x \; + \; 101
 

Columns :

(4 \, x^2 + 9 \, x + 2) \; + \; (3 \, x^2 + 5 \, x + 7) \; + \; (8 \, x^2 + x + 6)
= \; (2 \, x^2 + 9 \, x + 4) \; + \; (7 \, x^2 + 5 \, x + 3) \; + \; (6 \, x^2 + x + 8)
= \; 15 \, x^2 \; + \; 15 \, x \; + \; 15

(4 \, x^2 + 9 \, x + 2)^2 \; + \; (3 \, x^2 + 5 \, x + 7)^2 \; + \; (8 \, x^2 + x + 6)^2
= \; (2 \, x^2 + 9 \, x + 4)^2 \; + \; (7 \, x^2 + 5 \, x + 3)^2 \; + \; (6 \, x^2 + x + 8)^2
= \; 89 \, x^4 \; + \; 118 \, x^3 \; + \; 261 \, x^2 \; + \; 118 \, x \; + \; 89

 
and

{(4, 5, 6),   (2, 3, 1),   (9, 7, 8)}

(4 \, x^2 + 5 \, x + 6) \; + \; (2 \, x^2 + 3 \, x + 1) \; + \; (9 \, x^2 + 7 \, x + 8)
= \; (6 \, x^2 + 5 \, x + 4) \; + \; (x^2 + 3 \, x + 2) \; + \; (8 \, x^2 + 7 \, x + 9)
= \; 15 \, x^2 \; + \; 15 \, x \; + \; 15

(4 \, x^2 + 5 \, x + 6)^2 \; + \; (2 \, x^2 + 3 \, x + 1)^2 \; + \; (9 \, x^2 + 7 \, x + 8)^2
= \; (6 \, x^2 + 5 \, x + 4)^2 \; + \; (x^2 + 3 \, x + 2)^2 \; + \; (8 \, x^2 + 7 \, x + 9)^2
= \; 101 \, x^4 \; + \; 178 \, x^3 \; + \; 279 \, x^2 \; + \; 178 \, x \; + \; 101

 
{(8, 5, 2),   (1, 7, 4),   (6, 3, 9)}

(8 \, x^2 + 5 \, x + 2) \; + \; (x^2 + 7 \, x + 4) \; + \; (6 \, x^2 + 3 \, x + 9)
= \; (2 \, x^2 + 5 \, x + 8) \; + \; (4 \, x^2 + 7 \, x + 1) \; + \; (9 \, x^2 + 3 \, x + 6)
= \; 15 \, x^2 \; + \; 15 \, x \; + \; 15

(8 \, x^2 + 5 \, x + 2)^2 \; + \; (x^2 + 7 \, x + 4)^2 \; + \; (6 \, x^2 + 3 \, x + 9)^2
= \; (2 \, x^2 + 5 \, x + 8)^2 \; + \; (4 \, x^2 + 7 \, x + 1)^2 \; + \; (9 \, x^2 + 3 \, x + 6)^2
= \; 101 \, x^4 \; + \; 130 \, x^3 \; + \; 231 \, x^2 \; + \; 130 \, x \; + \; 101

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

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