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== {{main|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.<ref>{{citation|title=The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across the Dimensions|first=Clifford A.|last=Pickover|authorlink=Clifford A. Pickover|publisher=Princeton University Press|year=2011|isbn=9781400841516|page=7|url=https://books.google.com/books?id=-mu8O8RMG6QC&pg=PA7}}.</ref><ref>{{citation|title=Recreations in Mathematics|first=H. E.|last=Licks|publisher=D. Van Nostrand Company|year=1921|page=42|url=https://books.google.com/books?id=MWcxAQAAMAAJ&pg=PA42}}.</ref>
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.
111x121px|center
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:
<div class="center">111x121px111x121px</div>
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 :<math>\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}</math><ref>title=Determinant|url=https://mathworld.wolfram.com/Determinant.html</ref>
Here, <math>bfg, cdh, bdi,</math> and <math>afh</math> 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== {{reflist}}
{{numtheory-stub}} Category:Magic squares