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.
See main article: Pandiagonal magic square. 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 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.
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{align} |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{align}
Here,
bfg,cdh,bdi,
afh
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.