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This follows from the left side of the equation being equal to zero, requiring the right side to equal zero as well, and so the vector sum of a + b (the long diagonal of the rhombus) dotted with the vector difference a - b (the short diagonal of the rhombus) must equal zero, which indicates the diagonals are perpendicular.
The richer structure of geomagic squares is reflected in the existence of specimens showing a far greater degree of 'magic' than is possible with numerical types. Thus a panmagic square is one in which every diagonal, including the so-called broken diagonals, shares the same magic property as the rows and columns. However, it is easily shown ...
A pan-diagonal magic square remains a pan-diagonal magic square under cyclic shifting of rows or of columns or both. [69] This allows us to position a given number in any one of the n 2 cells of an n order square. Thus, for a given pan-magic square, there are n 2 equivalent pan-magic squares. In the example below, the original square on the ...
20+2+13+24+6 = 65 (center plus the remaining squares on its diagonals) Each of these quincunxes can be translated to other positions in the square by cyclic permutation of the rows and columns (wrapping around), which in a pandiagonal magic square does not affect the equality of the magic constants. This leads to 100 quincunx sums, including ...
A magic square in which the broken diagonals have the same sum as the rows, columns, and diagonals is called a pandiagonal magic square. [1] [2] Examples of broken diagonals from the number square in the image are as follows: 3,12,14,5; 10,1,7,16; 10,13,7,4; 15,8,2,9; 15,12,2,5; and 6,13,11,4. The fact that this square is a pandiagonal magic ...
A magic square is an arrangement of numbers in a square grid so that the sum of the numbers along every row, column, and diagonal is the same. Similarly, one may define a magic cube to be an arrangement of numbers in a cubical grid so that the sum of the numbers on the four space diagonals must be the same as the sum of the numbers in each row, each column, and each pillar.
A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix. The following matrix is square diagonal matrix: [] If the entries are real numbers or complex numbers, then it is a normal matrix as well. In the remainder of this article we will consider only square diagonal matrices, and refer to them ...
For the diagonal or pandiagonal classes, one or possibly 2 of the 6 oblique magic squares may be pandiagonal magic. All but 6 of the oblique squares are 'broken'. This is analogous to the broken diagonals in a pandiagonal magic square. i.e. Broken diagonals are 1-D in a 2-D square; broken oblique squares are 2-D in a 3-D cube.