Sesiano J. Magic Squares - Their History and Construction from Ancient Times to AD 1600

New York: Springer, 2019. — 321 p.The science of magic squares witnessed an important development in the Islamic world during the Middle Ages, with a great variety of construction methods being created and ameliorated. The initial step was the translation, in the ninth century, of an anonymous Greek text containing the description of certain highly developed arrangements, no doubt the culmination of ancient research on magic squares.CDefinitions
Particular cases of ordinary magic squares
Categories of order
Banal transformations of ordinary magic squares
Historical outline
Main sources considered
Squares transmitted to the Latin West
First attemptsDescriptionDiscovery of this method
Another way
A related method
Modifying the square’s aspect
A method brought from India
Separation by parity
Use of the knight’s move
Principles of these methods
The square of order
Method of dotting
Exchange of subsquares
Generalization
Continuous filling
Principles of these methods
Filling according to parity
An older method
Crossing the quadrants
Square of order 4
Squares of higher orders
Filling pairs of horizontal rows
Four knight’s routes
Knight’s shuttle
Filling by knight’s and bishop’s moves
Filling according to parity
Filling the subsquares of order
Exchanges in the natural square
An older method
Principles of these methods
Method of the cross
Construction of a border
Method of the central square
Composition using squares of orders larger than
Composition using squares of order
Examples
Cross in the middle
Central square
Preliminary observations
Empirical construction
Grouping the numbers by parity
Placing together consecutive numbers
Method of Stifel
Zigzag placing
Variation in the corner cells
Variation in the corner and middle cells
Equalization by means of the first numbers
Alternate placing
Method of Stifel
General principles of placing for even orders
Equalization by means of the first numbers
Cyclical placing
Method of Stifel
Principles of these methods
The main square and its parts
Filling the inner square
Filling the remainder of the square by trial and error
Completing the placing of odd numbers
Determining the number of cells remaining empty
Determining the sum required
Rules for placing the even numbers
First main equalization rule: excess or deficit of the form ±s, s >
Second main equalization rule: excess or deficit of the form ±(j−), j >
Neutral placings
First border
Other borders
Case of the order n = t + (with t > )
Other borders
The numbers form a single progression
The numbers form n progressions
Magic square with a set sum
Magic products
General observations
The given numbers are in the first row
The middle number is in the median lower cell
The given numbers are in the middle row
The given numbers are in the first row
The given numbers are in the diagonal
The given numbers are in opposite lateral rows
The given numbers are in the first two rows
General observations
The given numbers are in the upper row
The given numbers are in the second row
The given numbers are in the end cells of the first row and the median of the second
The given numbers are in the diagonal
The given numbers are in the corner cells
Writing in the sum as a whole
The given numbers are in the first row
The given numbers are in the first two rows, within the quadrants’ diagonals
The given numbers are in the first and third rows, within the diagonals
The given numbers are in the first and fourth rows, within the diagonals
Writing in the global sum
The given numbers are in the first row
The given numbers are equally distributed in the lateral rows
Squares of higher evenly-odd orders
Squares of odd orders
Squares of even orders
Squares of odd orders with central cell empty
Case of the square of order
Squares of evenly-even orders
Squares with divided cells
Magic triangles
Magic crosses
First type
Second type
Both sides are odd
One side evenly even and the other evenly odd
Both sides are evenly odd
Magic cubes
Appendices