An illustrated introduction to Caïssan squares: the magic of chess

We study various properties of n × n Caïssan magic squares. Following the seminal 1881 article by "Ursus" [Henry James Kesson (b. c. 1844)] in The Queen, we define a magic square to be Caïssan whenever it is pandiagonal and knight-Nasik so that all paths of length n by a chess bishop are magic (pandiagonal, Nasik, CSP1-magic) and by a (regular) chess knight are magic (CSP2-magic). We also study Caïssan beauties, which are pandiagonal and both CSP2- and CSP3-magic; a CSP3-path is by a special knight that leaps over 3 instead of 2 squares. Our paper ends with a bibliography of over 100 items (many with hyperlinks) listed chronologically from the 14th century onwards. We give special attention to items by (or connected with) "Ursus": Henry James Kesson (b. c. 1844), Andrew Hollingworth Frost (1819–1907), Charles Planck (1856–1935), and Pavle Bidev (1912–1988). We have tried to illustrate our findings as much as possible, and whenever feasible, with images of postage stamps or other philatelic items

トウサスウレツ オ フクム マホウジン

In this paper, we consider magic squares which contain arithmeticprogressions. Mainly, we consider the magic squares similar to Schaefer\u27s 7-squares. We show how to construct magic squares of order 4k+3 which areof strict Schaefer\u27s type. In general, it is unkown how to construct magicsquares of order 4k+1 which are of the type. However, we succeeded inconstructing magic squares of orders 9 and 13 which are of the type. At theend of this paper, we show an irregular pandiagonal magic 7-square of newclass

Polyhedral Cones of Magic Cubes and Squares

Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.Comment: 14 page

The Magic of Permutation Matrices: Categorizing, Counting and Eigenspectra of Magic Squares

Permutation matrices play an important role in understand the structure of magic squares. In this work, we use a class of symmetric permutation matrices than can be used to categorize magic squares. Many magic squares with a high degree of symmetry are studied, including classes that are generalizations of those categorized by Dudeney in 1917. We show that two classes of such magic squares are singular and the eigenspectra of such magic squares are highly structured. Lastly, we prove that natural magic squares of singly-even order of these classes do note exist.Comment: 26 page