Definition:Magic Square
Definition
A magic square is an arrangement of $n^2$ numbers into an $n \times n$ square array such that:
- the sum of the elements of each row
- the sum of the elements in each column
- the sum of the elements along each diagonal
are the same.
When the elements are not specified, it is conventional for them to be the first $n^2$ (strictly) positive integers.
Order
An $n \times n$ magic square is called an order $n$ magic square.
Magic Constant
The magic constant of a magic square is the number that each of the rows and columns adds up to.
Examples
Order $1$
The Order $1$ magic square is trivial:
- $\begin {array} {|c|} \hline 1 \\ \hline \end {array}$
Order $3$
- $\begin {array} {|c|c|c|} \hline 2 & 7 & 6 \\ \hline 9 & 5 & 1 \\ \hline 4 & 3 & 8 \\ \hline \end {array}$
Order $4$
There are many order $4$ magic squares.
Dürer's Order $4$ Magic Square
This is one of the more famous ones, due to Albrecht Dürer:
- $\begin {array} {|c|c|c|c|} \hline 16 & 3 & 2 & 13 \\ \hline 5 & 10 & 11 & 8 \\ \hline 9 & 6 & 7 & 12 \\ \hline 4 & 15 & 14 & 1 \\ \hline \end {array}$
Moessner's Order $4$ Magic Square
This one, created by Alfred Moessner, has extra interesting properties:
- $\begin {array} {|c|c|c|c|} \hline 12 & 13 & 1 & 8 \\ \hline 6 & 3 & 15 & 10 \\ \hline 7 & 2 & 14 & 11 \\ \hline 9 & 16 & 4 & 5 \\ \hline \end {array}$
Order $5$
- $\begin {array} {|c|c|c|c|c|} \hline 23 & 6 & 19 & 2 & 15 \\ \hline 10 & 18 & 1 & 14 & 22 \\ \hline 17 & 5 & 13 & 21 & 9 \\ \hline 4 & 12 & 25 & 8 & 16 \\ \hline 11 & 23 & 7 & 20 & 3 \\ \hline \end {array}$
Also known as
Some sources use the term magic square to what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is defined as a semi-magic square.
To such sources, a true magic square is then known as a diagonally magic square.
Hence it is wise to be careful to make sure exactly what is being referred to.
Also see
- Results about magic squares can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): magic square
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): magic square
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): magic square
- Weisstein, Eric W. "Magic Square." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MagicSquare.html