# Bent Diagonal Magic Squares

## Description

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 2. They are equivalent to U zigzagn/2. For odd order squares, the waves are equivalent to V zigzag(n+1)/2.

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.

### Bent n-Way

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.

## Odd Order

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 V zigzag 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.

## Even Order

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.

### Singly-Even

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:

 \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.

### Doubly-Even

#### Order 4

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.

#### Other Doubly-Even

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 V zigzag 2-way squares. The 2-way include inline and perpendicular, and some pandiagonal, compact, compact4, self-complement, U zigzag 2-way, V zigzag 2-way, and V zigzag3 2-way squares. The 4-way include some pandiagonal, compact, compact4, complete, Franklin, U zigzag, U zigzag 2-way, and self-complement squares.