In
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
and the theory of
magic squares, a broken diagonal is a set of ''n'' cells forming two
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...
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 A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the squ ...
.
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.
In linear algebra
Broken diagonals are used in a formula to find the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of 3 by 3
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
.
For a 3 × 3 matrix ''A'', its determinant is
:
[title=Determinant, url=https://mathworld.wolfram.com/Determinant.html]
Here,
and
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
{{numtheory-stub
Magic squares