![]() Thus in the above example for, the crossed-out numbers are originally 1, 4. Then replace each entry on a crossed-off diagonal by or, equivalently, reverse the order of the crossed-outĮntries. ![]() Subsquare and fill all squares in sequence. The second diagonal fills in 7, 9, 6, 8, 10, and so on.Īn elegant method for constructing magic squares of doubly even order In the above square, the first diagonal therefore fills in 1, 3, 5, 2, 4, Obtained by wrapping around the square until the wrapped diagonal reaches its initial That were missed are then added sequentially along the continuation of the diagonal Numbers are built up along diagonal lines in the shape of a diamond As illustrated above, in this method, the odd Sumdiffs for particular choices of ordinary and break vectors.Ī second method for generating magic squares of odd order has been discussed by J. H. Conway under the name of the "lozenge" If all sumdiffs are relativelyĪnd the square is a magic square, then the square is also a panmagic Call the set of these numbers the sumdiffs Magic squares can be constructed by considering the absolute sums, ,, and. Magic square, each break move must end up on an unfilled cell. Therefore has ordinary vector (1, and break vector (0, 1). That gives the offset to introduce upon a collision. To France after serving as ambassador to Siam.Ī generalization of this method uses an "ordinary vector" that gives the offset for each noncolliding move and a Is purported to have been first reported in the West when de la Loubere returned The method, also called de la Loubere's method, That is already filled, the next number is instead placed below the previous On the bottom and falling off the right returns on the left. The counting is wrapped around, so that falling off the top returns Row, then incrementally placing subsequent numbers in the square one unit above and It begins by placing a 1 in the center square of the top ![]() Technique known as the Siamese method can be used, as illustrated above (Kraitchikġ942, 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 square is said to be an associative magic If all pairs of numbers symmetrically opposite Produces another magic square, the square is said to be a bimagic 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 Pinn and Wieczerkowski (1998) Magic squares was computed by R. Schroeppel in 1973. The 880 squares of order four were enumerated by Frénicleĭe Bessy in 1693, and 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 rotationĢ.
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