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. A magic square in which the broken diagonals have the same sum as the rows, columns, and diagonals is called a panmagic square.^[1][2]

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.

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:

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.

References

1. Pickover, Clifford A. (2011), The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, p. 7, ISBN 9781400841516.
2. Licks, H. E. (1921), Recreations in Mathematics, D. Van Nostrand Company, p. 42.