The magic square contains magic patterns consisting of repeated waves.
These can be pan-zigzag, (i.e., including all in the direction of the bend),
up, down, left, and right.
On this site, *n*-way means **pan**-*n*-way.
By default, zigzag means pan-4-way; otherwise, it is qualified as 3-way, 2-way, or 1-way.

Zigzag squares are further classified as zigzag_{k}, for
k = 2,3,4,... where k refers to the number of cells
from trough to crest of a wave.

By default, zigzag refers to zigzag_{2}.
Zigzag_{2} squares can be only 2-way or 4-way.
See note.

Zigzag_{2}Squares
makes V zigzag_{2} squares for orders greater than 6. It uses two methods:

- Start with a V zigzag
_{2}center bones of order 7, 8, or 9. - Repeatedly add double borders of V zigzag
_{2}bones numbers.

- Start with an order 4 or a singly even order bordered magic square as center.
- Replace each number in the center square with a 2x2 block to make a double order square.
See notes.

Repeat, using each square as center, to continue doubling the order.

This method for order 8 was sent by Paul Michelet. Paul followed with an order 12 made by doubling a bordered order 6.

This method works for any doubly even order bigger than 4.

The squares are V zigzag_{k} and U zigzag_{k} for more values of k as the order
increases:

V zigzag_{k}U zigzag_{k}order k k, all squares k, some squares --------------- ----------- -------------- --------------- 8,12,20,.. 2 2 16,24,40,.. 2,3 2 4 32,48,80,.. 2,3,5 2,4 8 64,96,160, 2,3,5,9 2,4,8 16 128,192,320,.. 2,3,5,9,17 2,4,8,16 32 ... ... ... ...

Squares based on the 48 order 4 most-perfect squares are especially interesting.
These are somewhat similar to squares on Dwane and Keith Campbell's site,
(made by a different method).
They are also pandiagonal, complete, and
compact_{k} where k=n/2.
And they have
V zigzag_{k} pandiagonals:

order k ----- ----------- 8 2,3 16 2,3,5 32 2,3,5,9 64 2,3,5,9,17 ... ...

Note that some squares of doubly even order are also made by method I.

For 2-way left and right, the sum of numbers in alternating cells is the same in each
column. Let the average of the magic square numbers, (n^{2}+1)/2, be S1.
Then, for odd order, the sum of the alternating cell numbers starting in the top
row can be S1(n+1)/2; the sum of the alternating cell numbers starting in
the second row is then S1(n-1)/2.
For doubly even orders, the corresponding sums of alternating cell numbers are both
S1(n)/2. See notes.
There are no V zigzag_{2} squares of singly even order.

For 2-way up and down, these sums hold for the rows. For 4-way V zigzag_{2},
the sums hold for rows and columns:

There are 53568 order 5 V zigzag_{2} 2-way magic squares.
For odd orders greater than 5, there are V zigzag_{2} 2-way and
4-way magic squares.

There are no V zigzag magic squares of singly even order. Planck's proof, that there are no pandiagonal squares of singly even order, can be adapted for these squares. Substitute alternate zigzag paths for alternate diagonals in the proof.

There are 96 order 4 V zigzag_{2} 2-way magic squares.
These are **TYPE V** in the classification by Dudeney.

For doubly even orders greater than 4, there are V zigzag_{2} 2-way and
4-way magic squares.

Here are some V zigzag_{2}
8x8 squares.
They include V zigzag_{3} 2-way, U zigzag, and U zigzag 2-way,
adjacent corner, adjacent side, associative, pandiagonal complete, bent diagonal 2-way,
and compact_{4}.

Here are some V zigzag_{3} 1-way
8x8 squares.
They are also V zigzag 2-way and U zigzag 2-way.

Here are some V zigzag_{3} 2-way
8x8 squares.
They include V zigzag, V zigzag 2-way, U zigzag and U zigzag 2-way.

Here are some V zigzag_{3}
16x16 and
32x32
squares. They are also V zigzag and U zigzag.

The 16x16 are also U zigzag_{4} and include V zigzag_{5} 2-way, associative,
pandiagonal complete, bent diagonal 2-way, compact_{8}, and self-complement.

The 32x32 include U zigzag_{4}, U zigzag_{8}, associative, pandiagonal,
bent pandiagonal, compact_{8}, compact_{16}, and trimagic.

The magic constant for order 16 is 16x(16^{2}+1)/2 = 2056.

For the 16x16 squares, the sums of
numbers in every fourth cell is the same in each row and column: 2056/4 = 514.
So, the V zigzag_{3} and V zigzag paths are
all composed of 4 sums of 514.

One V zigzag_{3} right path is shown. One V zigzag down path is shown.

Note that 16 is an even multiple of the wavelength-1 for V zigzag and V zigzag_{3}.

The magic constant for order 32 is 32x(32^{2}+1)/2 = 16400.
For the 32x32 squares, the sums of
numbers in every fourth cell is the same in each row and column: 16400/4 = 4100.
So, the V zigzag_{3} and V zigzag paths are
all composed of 4 sums of 4100.

Here are some V zigzag_{4}
24x24 squares.
They are also V zigzag and U zigzag_{2,3,6}.

The magic constant for order 24 is 24x(24^{2}+1)/2 = 6924.

The sums of numbers in every sixth cell is the same in each row and column: 6924/6 = 1154.

So, the V zigzag_{4} and V zigzag paths are all composed of 6 sums of 1154.

For example, the first V zigzag_{4} down path sums are:

- row 1:
**1154 1154 1154 1154 1154 1154** - row 2:
**1154 1154 1154 1154 1154 1154** - row 3:
**1154 1154 1154 1154 1154 1154** - row 4:
**1154 1154 1154 1154 1154 1154** - ...

And, the first V zigzag down path sums are:

- row 1:
**1154 1154 1154 1154 1154 1154** - row 2:
**1154 1154 1154 1154 1154 1154** - ...

Here are some V zigzag_{5}
32x32 squares.
The first is from Dwane and Keith Campbell's site. It is also U zigzag 2-way, U zigzag_{4,8}, pandiagonal, bent diagonal, and compact_{16}.
The others are all also V zigzag, V zigzag_{3} and U zigzag_{4}. They
include V zigzag_{9} 2-way, U zigzag_{8}, U zigzag_{8} 2-way, associative,
bent diagonal 2-way, bimagic and bipandiagonal, pandiagonal, pandiagonal complete, compact_{16},
and self complement.

The magic constant for order 32 is 32x(32^{2}+1)/2 = 16400.

For some of the V zigzag_{2,3,5} squares, the sums of
numbers in every eighth cell is the same in each row and column: 16400/8 = 2050.
So, the V zigzag_{5}, V zigzag_{3}, and V zigzag paths are
all composed of 8 sums of 2050.

For other V zigzag_{2,3,5} squares, the sums are not 2050,
but, pairs of the sums, (1538 2562), total 4100. These pairs are distributed such that there are
4 of them in every V zigzag_{5}, V zigzag_{3}, and V zigzag path.

For one of these squares, the first V zigzag_{5} down path sums are:

- row 1:
**1538 1538 1538 1538 2562 2562 2562 2562** - row 2:
**1538 2562 2562 1538 2562 1538 1538 2562** - row 3:
**1538 2562 2562 1538 2562 1538 1538 2562** - row 4:
**1538 1538 1538 1538 2562 2562 2562 2562** - row 5:
**2562 2562 2562 2562 1538 1538 1538 1538** - ...

So, the first V zigzag_{3} down path sums are:

- row 1:
**1538 1538 1538 1538 2562 2562 2562 2562** - row 2:
**1538 2562 2562 1538 2562 1538 1538 2562** - row 3:
**1538 2562 2562 1538 2562 1538 1538 2562** - ...

And, the first V zigzag down path sums are:

- row 1:
**1538 1538 1538 1538 2562 2562 2562 2562** - row 2:
**1538 2562 2562 1538 2562 1538 1538 2562** - ...

For the first square, (V zigzag_{5} only), pairs of the sums, (1984 2116) and (1988 2112), total 4100.
These pairs are distributed such that there are
4 of them in every V zigzag_{5} path, but not in the V zigzag_{3} or V zigzag paths.

For example, the first V zigzag_{5} down and right path sums are:

- row 1:
**1984 1988 2112 2116 1984 1988 2112 2116** - row 2:
**2116 2112 1988 1984 2116 2112 1988 1984** - row 3:
**1988 1984 2116 2112 1988 1984 2116 2112** - row 4:
**2112 2116 1984 1988 2112 2116 1984 1988** - row 5:
**1984 1988 2112 2116 1984 1988 2112 2116** - ...

- col 1:
**1984 2116 1988 2112 1984 2116 1988 2112** - col 2:
**1988 2112 1984 2116 1988 2112 1984 2116** - col 3:
**2112 1988 2116 1984 2112 1988 2116 1984** - col 4:
**2116 1984 2112 1988 2116 1984 2112 1988** - col 5:
**1984 2116 1988 2112 1984 2116 1988 2112** - ...

So, the first V zigzag_{3} down path sums are:

- row 1:
**1984 1988 2112 2116 1984 1988 2112 2116** - row 2:
**2116 2112 1988 1984 2116 2112 1988 1984** - row 3:
**1988 1984 2116 2112 1988 1984 2116 2112** - ...

And, the first V zigzag down path sums are:

- row 1:
**1984 1988 2112 2116 1984 1988 2112 2116** - row 2:
**2116 2112 1988 1984 2116 2112 1988 1984** - ...

For wavelength 2, waves with bigger amplitudes can also be considered as V zigzag
if only the crest and trough cells are included in the magic pattern.
If a square is V zigzagA_{j} for a specific
value **j**, it is also V zigzagA_{k} for k = 2j-1, 3j-2,...
So, a square that is V zigzag is V zigzagA_{k} for k = 2,3,4,5,...,n.
See notes.

If a square is not V zigzag, it can be
V zigzagA_{j} for some value **j**.
For example, an order 8 square that is not V zigzag can be V zigzagA_{3}, i.e.,
V zigzagA_{3,5,7}. For odd orders, only V zigzag can be V zigzagA_{3}.
See note.

There are also V zigzagA squares for singly even orders. This was discovered when
Paul Michelet sent this
6x6
V zigzagA_{3} 2-way magic square, which he described as having "lateral knight moves".
Paul made this square as a composite of the 3x3 magic square and a selection of 2x2 squares.

A computer program was used to make some 6x6
V zigzagA_{3} 4-way magic squares.

In row 1, let **A** be the sum of the three cells labelled
**a**, and **B** the sum of the three
cells labelled **b**. If the square is magic, V zigzagA_{3}:

. the sum ofaaain each row and column isA. the sum ofbbbin each row and column isB.A+B= magic constant

For doubly even orders, if the square is also V Zigzag_{2}, **A** = **B**.
See notes.

There are 12 4x4
V zigzagA_{3} 2-way magic squares. They are bent diagonal 2-way, self-complement squares.

Here are some 6x6
V zigzagA_{3}, i.e., V zigzagA_{3,5}, magic squares.

Here are some 8x8
V zigzagA_{3} 2-way, i.e., V zigzagA_{3,5,7} 2-way, magic squares.

They include U zigzag 2-way, and bent diagonal 2-way squares.

Here are some 8x8
V zigzagA_{3}, i.e., V zigzagA_{3,5,7}, magic squares. They are also bimagic.

In this example of an order 8 V zigzagA_{5}, the sums of alternating cells in the
columns are all half the magic constant. Therefore, it is also V zigzag 2-way
(left, right). This is not generally the case. Some other known combinations of these sums
are:

- column (126,134)(134,126) row (126,134)(134,126)
- column (128,132)(132,128) row (126,134)(134,126)
- column (132,128)(128,132) row (130,130)
- column (134,126)(126,134) row (126,134)(134,126)

Here are some V zigzagA_{5}
8x8,
12x12,
16x16,
32x32
squares.

The 8x8 include V zigzag 2-way, U zigzag, U zigzag 2-way, associative,
adjacent corner, pandiagonal, bent diagonal, bent diagonal 1-way, bent diagonal 2-way,
and compact_{4} squares.

The 12x12 are V zigzagA_{5,9} and include
V zigzag 2-way, V zigzag_{4} 2-way, U zigzag,
U zigzag 2-way, U zigzag_{3}, U zigzag_{3} 2-way, adjacent corner,
associative, pandiagonal, pandiagonal 1-way, bent diagonal, bent diagonal 2-way, and
compact_{4} squares.

The 16x16 are V zigzagA_{5,9,13} and
include V zigzag 2-way, V zigzag_{3} 2-way,
U zigzag, U zigzag 2-way, U zigzag_{4}, U zigzag_{4} 2-way,
associative, pandiagonal, bent diagonal, bent diagonal 2-way, self-complement, compact_{4}, and
bimagic squares.

The 32x32 are V zigzagA_{5,9,13,17,21,25,29} and include
V zigzag_{5}, V zigzag_{5} 2-way, U zigzag 2-way,
U zigzag_{4,8}, pandiagonal, bent diagonal, bent diagonal 2-way, and
compact_{16} squares.

Campbell, Dwane H. and Keith A. Campbell "ORDER-2^{p} SQUARES"
http://magictesseract.com./large_squares

Dudeney, Henry E. "Magic Square Problems"

https://archive.org/details/AmusementsInMathematicspdf

Heinz, Harvey "Order 4 Magic Squares"

http://recmath.org/Magic%20Squares/order4list.htm#The 12 Groups

Michelet, Paul
Pimauchel@hotmail.com

A chess endgame and problem composer.
London, England.