Smallest Magic Square is of Order 3

Theorem
Apart from the trivial order $1$ magic square:

the smallest magic square is the order $3$ magic square:

Proof
Suppose there were an order $2$ magic square.

The row and column total is $\dfrac {1 + 2 + 3 + 4} 2 = 5$.

Any row or column with a $1$ in it must therefore also have a $4$ in it.

But there are:
 * one row
 * one column
 * one diagonal

which have a $1$ in it.

Therefore the $4$ would need to go in all 3 spaces.

But $4$ appears in a magic square exactly once.

Hence there can be no order $2$ magic square.