Showing papers on "Magic square published in 1987"

Merlin's magic square

Abstract: (1987) Merlin's Magic Square The American Mathematical Monthly: Vol 94, No 2, pp 143-150

Adjoint representations of exceptional Lie algebras

Abstract: The recent development of superstring theory (1) has drawn attention to the potential importance of the exceptional Lie algebras (particularly E/sub 6/ and E/sub 8/) in a future unified theory. The paper consists of three parts. In the first, the authors consider normed algebras and the properties of their transformations. In the second they give the construction of the magic square and the corresponding commutation relations. The construction is anticipated by two symmetric constructions, the symbiosis of which it is. In the final part, the exceptional algebras of the series E are decomposed. The proofs of the Jacobi identities and the identification of the algebras in the magic square are omitted since the calculations are lengthy.

Magic Squares and Magic Triangles

Abstract: For drill, many teachers use magic squares, η x η arrays of numbers in which the sums of the numbers in each row, each column, and each diagonal are the same. The only difficulty with magic squares is creating them. Once a specific magic square has been used, it isn't much fun to use it again. Students can follow clear-cut steps to create odd magic squares, that is, 3 x 3, 5 x 5, 7 x 7, and so on. Instead of stating the rules and regulations (which can be found in Fults [1974]) for creating such squares, a generic 3 x 3 magic square and a generic 5x5 magic square are presented in figure 1 . Any number can be used as the first number in an odd magic square. The only restriction in placing numbers in the magic square in the order indicated in figure 1 is that the "magic difference" must be constant, that is, any two succeeding numbers must have the same difference; for example, start with 1 and add 1 to get each succeeding number; start with 17 and subtract 3 to get each succeeding number. See the examples that appear in figure 2 (a 3 x 3 magic square). To create the 3 x 3 magic square starting with 0.15 and subtracting 0.06, 0.15 is placed as the first number (see fig. 1), then 0.15 0.06 = 0.09 is placed in the second position; 0.09 0.06 = 0.03 in the third position;