 ### Zigzag2 only 2-way or 4-way

Below, the letters a,b,c,d are only used to indicate zigzag paths. The letters do not represent cell values.

#### V Zigzag

Consider 2 rows:

```
a b a b a b a
c d c d c d c
```

If a zigzag up path cbcbcbc has magic sum, the zigzag down path adadada must also have magic sum; the two zigzag paths have the same total as the two rows.

#### U Zigzag

Consider 2 rows:

```
a b b a a b b a
c d d c c d d c
```

If a zigzag up path cbbccbbc has magic sum, the zigzag down path addaadda must also have magic sum; the two zigzag paths have the same total as the two rows. ### V ZigzagAk

Below, the letters a,b,c,d,... are only used to indicate zigzag paths. The letters do not represent cell values.

If a square is V zigzagAj for a specific value j, it is also V zigzagAk for k = 2j-1, 3j-2, ...

#### V ZigzagA2

A square that is V zigzag2 is V zigzagAk for k = 2,3,4,5,...,n.

Consider 3 rows:

```
a b a b a b a
c d c d c d c
e f e f e f e
```

Path adadada has magic sum and path cdcdcdc has magic sum. So, sum of a's = sum of c's. Path cfcfcfc has magic sum, so path afafafa has magic sum. So, V zigzag2 is V zigzagA3. Similarly, V zigzag2 is V zigzagAk for k = 4,5,...,n.

#### V ZigzagAj

If a square is V zigzagAj for a specific value j, it is also V zigzagAk for k = 2j-1, 3j-2, ...

Example: A square that is V zigzagA3 is V zigzagAk for k = 3,5,7.

Consider rows 1,3,5:

```
a b a b a b a
. . . . . . .
c d c d c d c
. . . . . . .
e f e f e f e
```

Path adadada has magic sum and path cdcdcdc has magic sum. So, sum of a's = sum of c's. Path cfcfcfc has magic sum, so path afafafa has magic sum. So, V zigzagA3 is V zigzagA5. Similarly, V zigzagA3 is V zigzagA7. #### V ZigzagA3 Property

Below, the letters a,b,c,d are only used to indicate zigzag paths. The letters do not represent cell values.

```Order 6 example:

a b a b a b
c d c d c d
a b a b a b
c d c d c d
a b a b a b
c d c d c d
```
```From above we know that:
the sum of the a's in each row,(Ar), is the same
the sum of the b's in each row,(Br), is the same
the sum of the c's in each row,(Cr), is the same
the sum of the d's in each row,(Dr), is the same
Similarly:
the sum of the a's in each column,(Ac), is the same
the sum of the b's in each column,(Bc), is the same
the sum of the c's in each column,(Cc), is the same
the sum of the d's in each column,(Dc), is the same

And 3Ar = 3Ac, so Ar = Ac. Similarly Br = Bc, Cr = Cc, Dr = Dc
3Ar + 3Br = 3Ac + 3Cc, so Cc = Br. ∴ Cr = Br.
3Ar + 3Br = 3Bc + 3Dc, so Dc = Ar. ∴ Dr = Ar.

a b a b a b           a b a b a b
c d c d c d           b a b a b a
So      a b a b a b    is     a b a b a b
c d c d c d           b a b a b a
a b a b a b           a b a b a b
c d c d c d           b a b a b a

With just 2 sums: A = Ar = Ac and B = Br = Bc

Note that for doubly even orders, if the square is also V zigzag2, A = B,
(because, for example, Br = Dr).
``` #### V ZigzagA3 must be V Zigzag2 for Odd Orders

Below, the letters a,b,c,d are only used to indicate zigzag paths. The letters do not represent cell values.

```Order 7 example:

a b a b a b a
c d c d c d c
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
e f e f e f e

From above we know that V ZigzagA3 is also V ZigzagA7, so:
sum of a's = sum of e's
sum of b's = sum of f's

And, from V ZigzagA3:
sum of e's = sum of c's
sum of f's = sum of d's

So:
sum of a's = sum of c's
sum of b's = sum of d's
``` ### Double Order Method

Order 8 example:

To make an 8x8 square, replace each number in a 4x4 square with a 2x2 block. The 4x4 can be any of the 712 4x4 squares that have diagonals made of complement pairs. The other 168 4x4 squares will make a semi-magic 8x8 square.

The 2x2 pattern can be any of:

```
1 4   1 4   2 3   3 2   3 2   4 1   4 1
2 3   3 2   1 4   1 4   4 1   2 3   3 2
```

Choose a pattern and use it for replacing the first half of the numbers, (1 to 8). Use the mirror aspect, rotate 4, of that pattern to replace the other half, (9 to 16). Example:

```
1  4  .  .  .  .  .  .
3  2  .  .  .  .  .  .
1 14 11  8         .  .  .  .  5  8  .  .
12  7  2 13         .  .  .  .  7  6  .  .
6  9 16  3         .  . 36 33  .  .  .  .
15  4  5 10         .  . 34 35  .  .  .  .
.  .  .  .  .  . 40 37
.  .  .  .  .  . 38 39

```

Repeat this process with the 8x8 square to make a 16x16 square and so on. ### V ZigZag Pandiagonals

It is easier to see these paths by extracting the pandiagonals as rows and applying the V zigzag pattern to these squares. Example: 