The magic square contains magic paths consisting of half-diagonal waves. These can be pan-bent-diagonal, (i.e., including all in the direction of the bend), up, down, left and right. On this site, n-way means pan-n-way unless otherwise stated. By default, bent-diagonal means pan-4-way; otherwise, it is qualified as 3-way, 2-way, or 1-way.
For even order squares, the waves are truncated, i.e., they contain a middle segment of height 0 and length 1. Even and odd 1-way patterns:
The commonly referred to bent diagonals are aligned at (n+1)/2 although they exist at other alignments. It is conceivable that these paths can also be pan-.. in the direction at right angles to the bend, i.e., horizontally in the above example.
For odd orders and doubly even orders, there are squares with pan- 1-way, 2-way, 3-way, and 4-way bent diagonals. The 2-way can be inline, for example, left and right; or, they can be perpendicular, for example, up and right.
Singly even orders are limited to 1-way and 2-way, and the 2-way can only be inline. See proof below.
For odd order squares, if these up bent wave paths are continued horizontally, the same alignment doesn't occur until after n - 1 repetitions. This shows the wave form at all horizontal alignments:
There are 9190 order 5 bent diagonal 1-way magic squares. There are also order 5 magic squares with 1-way bent diagonals at alignments 1, 2, 4 and 5 .
There are 21,446 order 7 bent diagonal 4-way magic squares.
Here are some order 7 bent diagonal 3-way, 2-way and 1-way magic squares.
The 2-way include inline and perpendicular, and some associative, pandiagonal, and zigzag2 2-way squares.
Here are some order 9 bent diagonal 4-way, 3-way, 2-way, and 1-way magic squares. The 2-way include inline and perpendicular.
For even order squares, if these up bent wave paths are continued horizontally, the pattern repeats with the same alignment. Of course, the wave form can be re-aligned to be pan- horizontally:
Note that if there is a pan- 2-way or 4-way at a given alignment, there will be the same pan- .. at the alignment plus n/2.
SelfComplementSquares makes bent diagonal 2-way magic squares for all even orders greater than 2.
For singly-even order there are bent diagonals 1-way and 2-way, (inline), but not more than 2-way. For more than 2-way, (or 2-way perpendicular), there have to be main bent diagonals in two perpendicular directions, say, up and right. Label the half main diagonals W, X, Y, Z and let the magic sum be Σ:
Back diagonal W + Z = Σ Up bent diagonal Y + Z = Σ So W = Y Right bent diagonal W + Y = Σ So W = Σ/2 But Σ/2 is fractional for singly even order.
Here are some order 6 bent diagonal 2-way and 1-way magic squares. The 2-way include some self-complement squares.
There are 304 order 4 bent diagonal 2-way magic squares. These are all side-to-side symmetric self-complement squares. At alignment 1, (or 3), there are another 96 order 4 bent diagonal 2-way magic squares. These are all adjacent side pair squares.
Harvey Heinz shows 48 order 4 bent diagonal squares. These are not pan-bent-diagonals, but there are 2 in each of the up, down, left and right directions. These are all adjacent corner pair squares.
Doubly even orders greater than 4 have magic squares with bent diagonals 4-way, 3-way, 2-way, and 1-way. Some are pandiagonal and some are most-perfect, for example, (Hurkens, Figure 14), .txt:
Except for 8k+4 orders, some of the 4-way are Franklin magic squares. Except for order 8, some are Franklin and most-perfect. This one is a variation of Hurkens, Figure 7: .txt.
Here are some order 8 bent diagonal 4-way, 3-way, 2-way, and 1-way magic squares. The 1-way include some adjacent side, and zigzag2 2-way squares. The 2-way include inline and perpendicular, and some pandiagonal, compact, self-complement, and zigzag2 2-way squares. The 4-way include some pandiagonal, compact, complete, Franklin, and self-complement squares.