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The first dateable instance of the fourth-order magic square occurred in 587 CE in India. Specimens of magic squares of order 3 to 9 appear in an encyclopedia from Baghdad c. 983, the Encyclopedia of the Brethren of Purity (Rasa'il Ikhwan al-Safa). By the end of 12th century, the general methods for constructing magic squares were well established.
Bi-Magic Squares of Order 9, 1999, 0-9684700-6-8 Curves and Approximations , 1999, 0-9684700-5-X An Inlaid Magic Tesseract , 1999, as a 17" x 22" poster OR an 8-page self-cover booklet Inlaid Magic Squares and Cubes (2nd edition), 2000, 0-9684700-3-3
A trimagic square is a magic square that remains magic when all of its numbers are replaced by their cubes. Trimagic squares of orders 12, 32, 64, 81 and 128 have been discovered so far; the only known trimagic square of order 12, given below, was found in June 2002 by German mathematician Walter Trump.
For example the following sequence can be used to form an order 3 magic square according to the Siamese method (9 boxes): 5, 10, 15, 20, 25, 30, 35, 40, 45 (the magic sum gives 75, for all rows, columns and diagonals). The magic sum in these cases will be the sum of the arithmetic progression used divided by the order of the magic square.
Bernard Frénicle de Bessy (c. 1604 – 1674), was a French mathematician born in Paris, who wrote numerous mathematical papers, mainly in number theory and combinatorics.He is best remembered for Des quarrez ou tables magiques, a treatise on magic squares published posthumously in 1693, in which he described all 880 essentially different normal magic squares of order 4.
As a running example, we consider a 10×10 magic square, where we have divided the square into four quarters. The quarter A contains a magic square of numbers from 1 to 25, B a magic square of numbers from 26 to 50, C a magic square of numbers from 51 to 75, and D a magic square of numbers from 76 to 100.
Apart from the trivial case of the first order square, most-perfect magic squares are all of order 4n. In their book, Kathleen Ollerenshaw and David S. Brée give a method of construction and enumeration of all most-perfect magic squares. They also show that there is a one-to-one correspondence between reversible squares and most-perfect magic ...
A pandiagonal magic square then would be a nasik square because 4 magic line pass through each of the m 2 cells. This was A.H. Frost’s original definition of nasik. A nasik magic cube would have 13 magic lines passing through each of its m 3 cells. (This cube also contains 9m pandiagonal magic squares of order m.)