Broken diagonal

In recreational mathematics and the theory of magic squares, a broken diagonal is a set of n cells forming two parallel diagonal lines in the square. Alternatively, these two lines can be thought of as wrapping around the boundaries of the square to form a single sequence.

In pandiagonal magic squares[]

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 square can be verified by checking that all of its broken diagonals add up to the same constant:

3+12+14+5 = 34

10+1+7+16 = 34

10+13+7+4 = 34

One way to visualize a broken diagonal is to imagine a "ghost image" of the panmagic square adjacent to the original:

The set of numbers {3, 12, 14, 5} of a broken diagonal, wrapped around the original square, can be seen starting with the first square of the ghost image and moving down to the left.

In linear algebra[]

Broken diagonals are used in a formula to find the determinant of 3 by 3 matrices.

For a 3 × 3 matrix A, its determinant is

{\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}\Box &\Box &\Box \\\Box &e&f\\\Box &h&i\end{vmatrix}}-b\,{\begin{vmatrix}\Box &\Box &\Box \\d&\Box &f\\g&\Box &i\end{vmatrix}}+c\,{\begin{vmatrix}\Box &\Box &\Box \\d&e&\Box \\g&h&\Box \end{vmatrix}}\\[3pt]&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}

^[3]

Here, $bfg,cdh,bdi,$ and $afh$ are (products of the elements of) the broken diagonals of the matrix.

Broken diagonals are used in the calculation of the determinants of all matrices of size 3 × 3 or larger. This can be shown by using the matrix's minors to calculate the determinant.

References[]

^ Pickover, Clifford A. (2011), The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across the Dimensions, Princeton University Press, p. 7, ISBN 9781400841516.
^ Licks, H. E. (1921), Recreations in Mathematics, D. Van Nostrand Company, p. 42.
^ title=Determinant|url=https://mathworld.wolfram.com/Determinant.html

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[1] Pickover, Clifford A. (2011), The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across the Dimensions, Princeton University Press, p. 7, ISBN 9781400841516.

[2] Licks, H. E. (1921), Recreations in Mathematics, D. Van Nostrand Company, p. 42.

[3] title=Determinant|url=https://mathworld.wolfram.com/Determinant.html

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