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,...,<=n. The k refers to the number of cells from trough to crest of a wave. Zigzag2 can be only 4-way or inline 2-way, (up, down or left, right).

Zigzag2

Construction

Zigzag2Squares makes zigzag2 squares for orders greater than 6. It uses the double border method:

Zigzag2 Property

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 can be both S1(n)/2. There are no zigzag2 squares of singly even order.

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

Odd Order

There are 53568 order 5 zigzag2 2-way magic squares.

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

Singly Even Order

There are no 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 zigzag2 2-way magic squares. These are TYPE V in the classification by Dudeney. See references.

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

Zigzag3

Here are some zigzag3 16x16 and 32x32 squares. They are also zigzag2.

The 16x16 include associative, pandiagonal complete, bent diagonal 2-way, self-complement, and zigzag5-2way.

The 32x32 include associative, pandiagonal, bent pandiagonal, 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 zigzag3 and zigzag2 paths are all composed of 4 sums of 514.

One zigzag3 right path is shown. Note that it includes 4 cells of each color.
One zigzag2 down path is shown. It too includes 4 cells of each color.

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 zigzag3 and zigzag2 paths are all composed of 4 sums of 4100.

Zigzag4

Here are some zigzag4 24x24 squares. They are also zigzag2.

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 zigzag4 and zigzag2 paths are all composed of 6 sums of 1154.
For example, the first zigzag4 down path sums are:

And, the first zigzag2 down path sums are:

Zigzag5

Here are some zigzag5 32x32 squares. The first is a pandiagonal, bent diagonal square from Dwane Campbell's site. The others are all also zigzag2 and zigzag3 and include associative, bent diagonal 2-way, bimagic and bipandiagonal, pandiagonal, pandiagonal complete, and self complement.

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

For some of the 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 zigzag5, zigzag3, and zigzag2 paths are all composed of 8 sums of 2050.

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

So, the first zigzag3 down path sums are:

And, the first zigzag2 down path sums are:

For the first square, (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 zigzag5 path, but not in the zigzag3 or zigzag2 paths.
For example, the first zigzag5 down and right path sums are:

So, the first zigzag3 down path sums are:

And, the first zigzag2 down path sums are:

ZigZagA

For wavelength 2, waves with bigger amplitudes can also be considered as zigzag if only the crest and trough cells are included in the magic pattern. If a square is zigzagAj for a specific value j, it is also zigzagAk for k = j, 2j-1, 3j-2,..., <= n. So, a square that is zigzag2 is zigzagAk for k = 2,3,4,5,..., <= n.

If a square is not zigzag2, it can be zigzagAj for some other value, j. This can happen when the order n is evenly divisible by j-1. For example, an order 16 square that is not zigzag2 can be zigzagA5. It is then also zigzagA9 and zigzagA13.

ZigzagA3

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

Here are some 8x8 zigzagA3 2-way magic squares. They include bent diagonal 2-way squares.

ZigzagA5

In this example of an order 8 zigzagA5, the sums of alternating cells in the columns are all half the magic constant. Therefore, it is also zigzag2 2-way (left, right). This is not generally the case. Some other known combinations of these sums are:

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

The 8x8 include adjacent corner, pandiagonal, bent diagonal, bent diagonal 2-way, and zigzag2 2-way squares.

The 12x12 include adjacent corner, associative, pandiagonal, pandiagonal 1-way, bent diagonal, bent diagonal 2-way, zigzag2 2-way, and zigzag4 2-way squares.

The 16x16 include associative, pandiagonal, bent diagonal, bent diagonal 2-way, zigzag2 2-way, zigzag2,3 2-way, self-complement, and bimagic squares.

The 32x32 include pandiagonal, bent diagonal, bent diagonal 2-way, zigzag5, and zigzag5 2-way squares.

REFERENCES

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

Dudeney, Henry E. "Magic Square Problems"
http://www.web-books.com/Classics/Books/B0/B873/AmuseMathC14P1.htm

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