Magic Constant of Order 4 Magic Square

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): $34$