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Sacred Number

52 53 furTher MagiC sQuares A magic square is normal if it uses whole numbers from 1 to the square of its order, and simple if its only property is rows, columns and main diagonals adding to the magic sum. The normal magic square of order3 is unique apart from 8 possible reflections and rotations or aspects 4 below. If the numbers in a magic square sum symmetrically about the center, e.g. 28, 73... the square is associated not simple, the number pairs are complementary. There are 880 order4 normal magic squares. To count magic squares mathematicians rotatereflect them so the topleft cell is as small as possible with the cell to its right less than the cell below. Complementary numbers in normal order4 squares form 12 Dudeney patterns 4 shown below. The 48 Group I squares are pandiagonal, the 6 broken diagonals formed by opposite sides wrapping round to meet each other also sum magically below left and center. Order4 pandiagonal magic squares are also mostperfect, any 2by2 square, including wrap arounds, adds to the magic sum above right. Only normal pandiagonal squares of doublyeven order 4, 8, 12 can be mostperfect. There are 275,305,224 normal order5 magic squares. Order5 is the lowest order of magic squares that can be pandiagonal and associated at the same time one shown here. There are 36 essentially different pandiagonal order5 magic squares, each produces 99 variations by permuting rows, columns and diagonals for a total of 3,600 pandiagonal order5 squares. It is not known how many normal order6 magic squares there are. Order6 is the first oddlyeven order, divisible by 2 but not by 4, the hardest squares to construct. It is impossible for a normal order6 square to be pandiagonal or associated. To construct a magic square of doublyeven order, place the numbers in sequence from top left as below. Using the pattern shown exchange every number on a marked diagonal with its complement and you have a magic square. To make a magic square of any odd order place 1 in the top middle cell and place numbers in sequence up and to the right by one cell, wrapping topbottom and rightleft as necessary. When a previously filled cell is reached move down one cell instead. The central cell will contain the middle number of the sequence and the diagonals will add to the magic sum alternative fill pattern below right. Two magic squares combine to make a composition magic square with the original orders multiplied together. Make copies of the first square left as if each were a cell in the second square center. Subtract 1 from each cell in the second square and multiply by the number of cells in the first square right. Add these to each cell in the large square. To make a bordered magic square add double the order, plus 2 to the cells of a normal magic square and make a border of the highestlowest numbers in the new sequence. A magic square within another that doesnt follow the highestlowest number border rule is an inlaid magic square. Also possible are inlaid magic diamonds and embedded magic squares orders 3 4 in order7 below. A bimagic square is still magic if all its numbers are squared. This one has a magic sum of 369. Each 3by3 section also has this sum. The squared magic sum is 20,049. In 3 dimensions we find the surprising possibility of
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