You might now be wondering whether there is an easy way to make a magic square without resorting to guesswork. Nobody knows how many distinct magic squares exist of order 6, but it is estimated to be more than a million million million! De La Loubere and the Siamese Method There are 880 distinct magic squares of order 4 and 275,305,224 of order 5.
Counted in this way, there is only one magic square of order 3, which is the Lo Shu magic square shown above. Mathematicians normally regard two magic squares as being the same if you can obtain one from the other by rotation or There's only one magic square of order 1 and it isn't particularly interesting: a single square with the number 1 inside! You can work out for yourself why the square of order 2 does not exist. It turns out that normal magic squares exist for all orders, except order 2. But since every number between 1 and n 2 appears exactly once in the square, you know that the total number is also equal toĪnd, as many of you may know, this sum is equal to n 2(n 2+1)/2. So nM(n) is the value you get when you add up all the entries in the square. It is easy to derive this formula: a magic square of order n has exactly n rows, and each row adds up to the magic constant M(n). For a magic square of order n, the magic constant is
We call this number the magic constant, and there's a simple formula you can use to work out the magic constant for any normal magic square. In the Lo Shu magic square, which is a normal magic square, all the rows, all the columns and the two diagonals add up to the same number, 15. Not surprisingly, magic squares made in this way are called normal magic squares. For example, a magic square of order 3 contains all the numbers from 1 to 9, and a square of order 4 contains the numbers 1 to 16. In a typical magic square, you start with 1 and then go through the whole numbers one by one. For example, a 3 by 3 magic square has three rows and three columns, so its order is 3. This is just the number of rows or columns that the magic square has. When mathematicians talk about magic squares, they often talk about the order of the square.