Broken diagonal

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A broken diagonal is a sequence of n diametrically-positioned cells in a panmagic square which spans the vertical boundary. Examples of broken diagonals from the below square 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.

PanmagicSquare-Order4.png

Notice that because one of the properties of a panmagic square is that the broken diagonals add up to the same constant, the following pattern is evident:

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:

PanmagicSquare-Order4.pngPanmagicSquare-Order4.png

It is easy to see now how the set of numbers {3, 12, 14, 5} result to form a broken diagonal: once wrapped around the original square, it can now be seen starting with the first square of the ghost image and moving down to the left.

Although this specific article relates to broken diagonals in panmagic squares, the "broken diagonal" also has uses in matrices and other areas of geometry such as determining the area of a polygon solely from its Cartesian coordinates.