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

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.

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.

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 zigzagA_{j} for a specific value j, it is also V zigzagA_{k}
for k = 2j-1, 3j-2, ...

A square that is V zigzag_{2} is V zigzagA_{k} 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 zigzag_{2} is V zigzagA_{3}.
Similarly, V zigzag_{2} is V zigzagA_{k} for k = 4,5,...,n.

If a square is V zigzagA_{j} for a specific value j, it is also V zigzagA_{k}
for k = 2j-1, 3j-2, ...

Example:
A square that is V zigzagA_{3} is V zigzagA_{k} 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 zigzagA_{3} is V zigzagA_{5}. Similarly, V zigzagA_{3} is V zigzagA_{7}.

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 thea's in each row,(A_{r}), is the same the sum of theb's in each row,(B_{r}), is the same the sum of thec's in each row,(C_{r}), is the same the sum of thed's in each row,(D_{r}), is the same Similarly: the sum of thea's in each column,(A_{c}), is the same the sum of theb's in each column,(B_{c}), is the same the sum of thec's in each column,(C_{c}), is the same the sum of thed's in each column,(D_{c}), is the same And 3A_{r}= 3A_{c}, soA_{r}=A_{c}. SimilarlyB_{r}=B_{c},C_{r}=C_{c},D_{r}=D_{c}3A_{r}+ 3B_{r}= 3A_{c}+ 3C_{c}, soC_{c}=B_{r}. ∴C_{r}=B_{r}. 3A_{r}+ 3B_{r}= 3B_{c}+ 3D_{c}, soD_{c}=A_{r}. ∴D_{r}=A_{r}.a b a b a b a b a b a b c d c d c d b a b a b aSoa b a b a bisa b a b a bc 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 aWith just 2 sums:A=A_{r}=A_{c}andB=B_{r}=B_{c}Note that for doubly even orders, if the square is also V zigzag_{2},A=B, (because, for example,B_{r}=D_{r}).

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 eFrom above we know that V ZigzagA_{3}is also V ZigzagA_{7}, so: sum of a's = sum of e's sum of b's = sum of f's And, from V ZigzagA_{3}: 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

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.

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