8...1...6
3...5...7
4...9...2
Step-by-step explanation:
Magic squares, those seemingly innocent looking collections of numbers that have fascinated so many for centuries, were known to the ancients, and were thought to possess mystical qualities and magical powers because of their unusual nature. In reality, they are nor=t as magic as they are fascinating since they are usually created by following a specific set of rules or guidelines. Their creation has been a constant source of amusement for many over the years as well as studying them for their seemingly mystical properties. History records their presence in China prior to the Christian era and their introduction into Europe is believed to have occurred in the 15th century. The study of the mathematical theory behind them was initiated in France in the 17th century and subsequently explored in many other countries.
Most people are quite familiar with the basic, and traditional, magic square where the sum of each row, column, and main diagonals, add up to a constant. The basic magic square of order n, that is n rows and n columns, or an n x n array, uses the integers from 1 to n^2. A magic square is usually referred to as a 3 cell, 4 cell, 5 cell, etc., or as a 3x3 array, 4x4 array, 5x5 array, etc. The basic square is made up of the consecutive integers from 1 to n^2 with all rows, columns, and long diagonals adding up to a constant C that is defined by
C = n(n^2 + 1)/2
The integers do not have to start with one nor do they have to be truly consecutive. A magic square can start with any number and the difference between the successive set of numbers can be any common difference, that is, in arithmetic series. For instance, it could start with 15 and progress with a common difference between successive numbers of say 3. The new constant C for this type of magic square can be derived from
C = n[2A + D(n^2 - 1)]/2
where A is the starting integer, D is the common difference between successive terms, and n is as defined earlier.
Obviously, an infinite number of squares can be made using these open boundries and rules. Consider also the squares that can be created by rotating and reflecting the basic squares and those not starting with 1. Considering only the basic squares starting with 1, there is only one 3rd order magic square. There are 880 different 4th order basic squares and approximately 320,000,000 different 5th order basic squares. Wow!
Would you believe that it is possible to create a magic square where every row, column, and main diagonal, add up to a different number?
There are many other types of magic squares. A magic square where one, or both, of the main diagonal sums is different from the rectangular sums, is called a semi-magic square. Squares where all the diagonal sums are equal to the all the rectangular sums are called panmagic squares. A square created by replacing each of its numbers by its square is referred to as bimagic while one created by replacing each of its numbers by its cubes is called trimagic.
There are an unlimited number of orders for squares, hundreds of different methods of forming squares, and countless rotations and reflections of the squares. There are odd and even order squares, doubly even squares, bordered squares, symmetrical squares, pandiagonal squares, non-consecutive squares, trebly magic squares, etc., all created by Strachey's rule, De la Loubere's rule, Arnoux's rule, Margossian's method, Plank's method, Kraitchik's method, Heath's method, etc., and the list goes on and on. Only the simplest are discussed below.
ODD CELL SQUARES
The simplest magic square has 3 cells on a side, or 9 cells altogether. We call this a three square. The simplest three square is one where you place the numbers from 1 to 9, inclusive, in each cell in such a way that the sum of every horizontal and vertical row as well as the two diagonal rows add up to the same number. This basic square looks like the following:
8 1 6
3 5 7
4 9 2
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