Zigzag Magic Squares

Description

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 zigzagk, 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 zigzag2. Zigzag2 squares can be only 2-way or 4-way. See note.

V Zigzag

U Zigzag

Construction

Zigzag2Squares makes V zigzag2 squares for orders greater than 6. It uses two methods:

I: Double Border Method

II: Double Order Method

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 zigzagk and U zigzagk for more values of k as the order increases:


                       V zigzagk                    U zigzagk 
       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 compactk where k=n/2. And they have V zigzagk 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.

V Zigzag2

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, (n2+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 zigzag2 squares of singly even order.

For 2-way up and down, these sums hold for the rows. For 4-way V zigzag2, the sums hold for rows and columns:

Odd Order

There are 53568 order 5 V zigzag2 2-way magic squares. For odd orders greater than 5, there are V zigzag2 2-way and 4-way magic squares.

Singly Even Order

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.

Doubly Even Order

There are 96 order 4 V zigzag2 2-way magic squares. These are TYPE V in the classification by Dudeney.

For doubly even orders greater than 4, there are V zigzag2 2-way and 4-way magic squares.

Here are some V zigzag2 8x8 squares. They include V zigzag3 2-way, U zigzag, and U zigzag 2-way, adjacent corner, adjacent side, associative, pandiagonal complete, bent diagonal 2-way, and compact4.

V Zigzag3

Here are some V zigzag3 1-way 8x8 squares. They are also V zigzag 2-way and U zigzag 2-way.

Here are some V zigzag3 2-way 8x8 squares. They include V zigzag, V zigzag 2-way, U zigzag and U zigzag 2-way.

Here are some V zigzag3 16x16 and 32x32 squares. They are also V zigzag and U zigzag.

The 16x16 are also U zigzag4 and include V zigzag5 2-way, associative, pandiagonal complete, bent diagonal 2-way, compact8, and self-complement.

The 32x32 include U zigzag4, U zigzag8, associative, pandiagonal, bent pandiagonal, compact8, compact16, and trimagic.

The magic constant for order 16 is 16x(162+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 zigzag3 and V zigzag paths are all composed of 4 sums of 514.

One V zigzag3 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 zigzag3.

The magic constant for order 32 is 32x(322+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 zigzag3 and V zigzag paths are all composed of 4 sums of 4100.

V Zigzag4

Here are some V zigzag4 24x24 squares. They are also V zigzag and U zigzag2,3,6.

The magic constant for order 24 is 24x(242+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 zigzag4 and V zigzag paths are all composed of 6 sums of 1154.
For example, the first V zigzag4 down path sums are:

And, the first V zigzag down path sums are:

V Zigzag5

Here are some V zigzag5 32x32 squares. The first is from Dwane and Keith Campbell's site. It is also U zigzag 2-way, U zigzag4,8, pandiagonal, bent diagonal, and compact16. The others are all also V zigzag, V zigzag3 and U zigzag4. They include V zigzag9 2-way, U zigzag8, U zigzag8 2-way, associative, bent diagonal 2-way, bimagic and bipandiagonal, pandiagonal, pandiagonal complete, compact16, and self complement.

The magic constant for order 32 is 32x(322+1)/2 = 16400.

For some of the V zigzag2,3,5 squares, the sums of numbers in every eighth cell is the same in each row and column: 16400/8 = 2050. So, the V zigzag5, V zigzag3, and V zigzag paths are all composed of 8 sums of 2050.

For other V zigzag2,3,5 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 zigzag5, V zigzag3, and V zigzag path.
For one of these squares, the first V zigzag5 down path sums are:

So, the first V zigzag3 down path sums are:

And, the first V zigzag down path sums are:

For the first square, (V zigzag5 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 zigzag5 path, but not in the V zigzag3 or V zigzag paths.
For example, the first V zigzag5 down and right path sums are:

So, the first V zigzag3 down path sums are:

And, the first V zigzag down path sums are:

V ZigzagA

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 zigzagAj for a specific value j, it is also V zigzagAk for k = 2j-1, 3j-2,... So, a square that is V zigzag is V zigzagAk for k = 2,3,4,5,...,n. See notes.

If a square is not V zigzag, it can be V zigzagAj for some value j. For example, an order 8 square that is not V zigzag can be V zigzagA3, i.e., V zigzagA3,5,7. For odd orders, only V zigzag can be V zigzagA3. See note.

There are also V zigzagA squares for singly even orders. This was discovered when Paul Michelet sent this 6x6 V zigzagA3 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 zigzagA3 4-way magic squares.

V ZigzagA3, (Knight Move), Property

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 zigzagA3:


        . the sum of aaa in each row and column is A
        . the sum of bbb in each row and column is B
        . A + B = magic constant

For doubly even orders, if the square is also V Zigzag2, A = B. See notes.

V ZigzagA3 Examples

There are 12 4x4 V zigzagA3 2-way magic squares. They are bent diagonal 2-way, self-complement squares.

Here are some 6x6 V zigzagA3, i.e., V zigzagA3,5, magic squares.

Here are some 8x8 V zigzagA3 2-way, i.e., V zigzagA3,5,7 2-way, magic squares.
They include U zigzag 2-way, and bent diagonal 2-way squares.

Here are some 8x8 V zigzagA3, i.e., V zigzagA3,5,7, magic squares. They are also bimagic.

V ZigzagA5

In this example of an order 8 V zigzagA5, 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:

Here are some V zigzagA5 8x8, 12x12, 16x16, 32x32 squares.

The 8x8 include V zigzag 2-way, U zigzag, U zigzag 2-way, adjacent corner, pandiagonal, bent diagonal, bent diagonal 1-way, bent diagonal 2-way, and compact4 squares.

The 12x12 are V zigzagA5,9 and include V zigzag 2-way, V zigzag4 2-way, U zigzag, U zigzag 2-way, U zigzag3, U zigzag3 2-way, adjacent corner, associative, pandiagonal, pandiagonal 1-way, bent diagonal, bent diagonal 2-way, and compact4 squares.

The 16x16 are V zigzagA5,9,13 and include V zigzag 2-way, V zigzag3 2-way, U zigzag, U zigzag 2-way, U zigzag4, U zigzag4 2-way, associative, pandiagonal, bent diagonal, bent diagonal 2-way, self-complement, compact4, and bimagic squares.

The 32x32 are V zigzagA5,9,13,17,21,25,29 and include V zigzag5, V zigzag5 2-way, U zigzag 2-way, U zigzag4,8, pandiagonal, bent diagonal, bent diagonal 2-way, and compact16 squares.

REFERENCES

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

Dudeney, Henry E. "Magic Square Problems"
https://archive.org/details/AmusementsInMathematicspdf

Heinz, Harvey "Order 4 Magic Squares"
http://www.magic-squares.net/order4list.htm#The 12 Groups

Michelet, Paul    Pimauchel@hotmail.com
A chess endgame and problem composer. London, England.