Magic Squares
If we draw a 3x3 grid (three squares per Row & three squares per Column), we will obtain a total of nine (9) squares overall in
which we can arrange numbers in different combinations. If we consider at first (because of the total number of squares in the
grid and because this is the series of numbers consisting the basis of the Decimal numerical system) arranging in the grid the
nine first Natural Numbers 1…9, we can obtain something similar to the following:
If we now rearrange the numbers in the grid with the purpose of assigning to the a specific property and not just have them the
one after the other as they appear in the set of Natural Numbers we can obtain something similar to the following:
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The above presented Square of Numbers is then a 3x3 Magic Square. An obvious question arising straight after this statement is:
“and what’s so Magic about it?” In order to verify the Magic properties of this Square of Numbers we will need to only perform
a few additions. If we add the three numbers in the first Row of the grid, we will find out that the total is 15. If we add the
three numbers in the second Row of the grid, we will find out that the total is 15. If we add the three numbers in the third Row
of the grid, we will find out that the total is 15. If we add the three numbers in the first Column of the grid, we will find out
that the total is 15. If we add the three numbers in the second Column of the grid, we will find out that the total is 15. If we
add the three numbers in the third Column of the grid, we will find out that the total is 15. If we add the three numbers in the
first Diagonal of the grid, we will find out that the total is 15, and finally, if we add the three numbers in the second Diagonal
of the grid, we will find out that the total is 15. The concept is summarised in the figure below.
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The number 15 is called the “Magic Sum” or “Magic Constant” of the Square. A Magic Square that contains the integers from 1 to n^2
is called a “Normal Magic Square” (the numbers making up the Magic Square are consecutive whole numbers starting from the number 1).
We can construct Normal Magic Squares of all sizes apart from 2x2 (that is, where n = 2). The 1x1 Magic Square, with only one cell
that contains number 1 only, is considered to be trivial. Effectively this makes the 3x3 Magic Square to be the smallest non-trivial
we can construct.
Any Magic Square can be “Rotated” and “Reflected” to produce another eight (8) arrangements giving the same Magic Sum. In Magic Square
Theory all of these combinations are generally deemed equivalent and all nine (9) combinations are taken as constituting a single
“Equivalence Class”.
Every Normal Magic Square has a Constant dependent on n, calculated by the formula: M = [n(n^2 + 1)] / 2. For Normal Magic Squares of
Order n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175 & 260.
A 3x3 Magic Square can be increased in size and transformed in a 4x4 Magic Square. With a 4x4 Magic Square there are more than one ways
of constructing them by using the first 16 numbers from the Natural Numbers Set (N) : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16. Two examples of these different arrangements are given below:
The second Magic Square happens to be the very famous 4x4 Magic Square that appears in an engraving entitled “Melancholia”
by the German painter and sculptor Albrecht Durer. It was made in 1514. This Magic Square is so “Magic” that even has the
year number 1514 constructed by its two middle numbers in the very bottom Row.
Among all different arrangements for the 4x4 Magic Square, there is one that possesses more properties than the eye can meet
at first. This arrangement is the following:
This arrangement of the 4x4 Magic Square possesses mainly the addition property in respect to all others that we can add up
its numbers in much more different ways than just the Rows, Columns & Main Diagonals, and still get the sum (34). Of course
we do not just select random numbers from the Magic Square, neither “convenient” numbers. We always follow a specific pattern
in our grouping of numbers:
At the flip-side of the coin lie Magic Squares which hold the special property of having none of their Rows, Columns &
Main Diagonals adding up to the same number. As surprising as it may seem at first, this proves to be as difficult if
attempted to achieve as trying to formulate a Magic Square. Charles Trigg was among the first to succeed in constructing
one such Magic Square and it became so popular that this kind of Magic Squares acquired a special name. They are now called
“HeteroSquares”. The following is Charles Trigg’s 3x3 HeteroSquare:
Taking the concept a step further, the following is a characteristic example of a Perfect 5x5 Magic Square:
As expected there are much more and quite new adding patterns to explore in respect to a 5x5 Magic Square than
exist with lower resolution Magic Squares. The “Centre Start” and the “Corner Start” adding patterns, following,
are only two examples:
Complementary to the Star-like patterns are other patterns like the “Rhomboid” ones. The fundamental rule in testing
for Rhomboid patterns is that every valid Rhomboid pattern must necessarily include the number at the centre of the
Magic Square, like those in the example figures below:
An extension of a 5x5 Magic Square can be considered the situation at which we have a 3x3 Magic Square contained inside it.
The 5x5 Magic Square can even be a Pure Magic Square with all its Rows, Columns, and Main Diagonals complying with
requirements. The 3x3 Magic Square now, contained in it (usually lying at its centre but not necessarily) is also a Magic
Square, but not a Pure Magic Square since usually numbers other than 1, 2, 3, 4, 5, 6, 7, 8, and 9 is required to be used
for its construction. However, at least for most of the cases is useful to have in mind that numbers used in the 3x3 Magic
Square is proven to be consecutive, “magically” starting with 9 and ending with 16:
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If any suspicion has been arisen that far that researching and creating Magic Squares is a trivial mathematical exercise with
no real applications that only middle-skilled unknown mathematicians ever took over, the following couple of names will server
into putting this right. Benjamin Franklin invented the following 8x8 Magic Square in his spare time and is a Pure Magic Square
because consecutive counting numbers have been used from 1 to 64. He even got so fascinated by Magic Squares and their “magic”
properties that he also proceeded into developing a 16x16 Magic Square as an extension to the one presented below with two
8-number patterns highlighted:
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The French mathematician Bernard de Bessy in the 1600s investigated the possibilities of having Magic Squares nested within
Magic Squares. From his work one Magic Square really stands out and is the one presented below. The arrangement is as simple
as it is effective in demonstrating pure “Mathematical Magic”. A 9x9 Magic Square which contains: a 3x3 Magic Square, a 5x5
Magic Square, and a 7x7 Magic Square, respectively. A closer look will easily reveal the fact that the gradually smaller
Magic Squares are not Pure Magic Squares, but this does not loosen the grip of its “Magic Spell” the least!
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