The standard Siamese method therefore has The offset to introduce upon a collision. The offset for each noncolliding move and a "break vector" that gives When de la Loubere returned to France after serving as ambassador to Siam.Ī generalization of this method uses an "ordinary vector" that gives The method, alsoĬalled de la Loubere's method, is purported to have been first reported in the West When a square is encountered that is already filled, the next number is instead placedīelow the previous one and the method continues as before. Off the top returns on the bottom and falling off the right returns on the left. The counting is wrapped around, so that falling Square of the top row, then incrementally placing subsequent numbers in the square For odd,Ī very straightforward technique known as the Siamese method can be used, as illustratedĪbove (Kraitchik 1942, pp. 148-149). Kraitchik (1942) gives general techniques of constructing even and odd squares of order. In addition, squares that are magic under both addition and multiplicationĬan be constructed and are known as addition-multiplication Squares that are magic under multiplication instead of addition can be constructed and are known as multiplication magic squares. The center sum to, the square is said to be an associative magic square. If all pairs of numbers symmetrically opposite If a square is magic for, , and, it is called a trimagic Magic square, the square is said to be a bimagic square Square (also called a diabolic square or pandiagonal square). Those obtained by wrapping around) of a magic square sum to the magicĬonstant, the square is said to be a panmagic (1982) and onĪ square that fails to be magic only because one or both of the main diagonal sums do not equal the magic constant is called a semimagic square. Methods forĮnumerating magic squares are discussed by Berlekamp et al. Using Monte Carlo simulation and methods from statistical mechanics. The number of squares is not known, but PinnĪnd Wieczerkowski (1998) estimated it to be Of magic squares was computed by R. Schroeppel The 880 squares of order four were enumerated by Frénicle de Bessy in 1693,Īnd are illustrated in Berlekamp et al. It is an unsolved problem to determine the number of magic squares of an arbitrary order, but the number of distinct magic squares (excluding those obtained by rotationĪnd reflection) of order, 2.