Magic Constant of Order 4 Magic Square
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Theorem
The magic constant of the order $4$ magic square is $34$.
Proof 1
Let $M_4$ denote an order $4$ magic square
By Sum of Terms of Magic Square, the total of all the entries in $M_4$ is given by:
- $T_4 = \dfrac {4^2 \left({4^2 + 1}\right)} 2 = \dfrac {16 \times 17} 2 = 136$
As there are $4$ rows of $M_4$, the magic constant of $M_4$ is given by:
- $S_4 = \dfrac {136} 4 = 34$
$\blacksquare$
Proof 2
Let $M_n$ denote the magic square of order $n$.
By Magic Constant of Magic Square, the magic constant of $M_n$ is given by:
- $S_n = \dfrac {n \left({n^2 + 1}\right)} 2$
Setting $n = 4$:
- $S_4 = \dfrac {4 \times 17} 2 = 34$
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $34$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $34$