mathematics Science and Technology

Recreational Math I: Magic Squares: the “really good” kind – Part 6

I have met with some disappointment as to how a methodology for creating a 4×4 square should pan out, and instead I have come up with many different algorithms, each resulting in its own small sets of magic squares, but had stumbled upon a set of squares with similar “hyper-magical” properties which I called the Durer Series.

I had met with some disappointment at this, and am still in the middle of writing my own 4×4 square program in Visual Basic .NET (it’s going to use a “brute force” algorithm … sorry!). With that, I had to learn about how .NET does objects. To those of you out of the loop on the recent .NET versions of VB, this language actually allows you to create your own novel objects, thus saving processing time if the right kind of objects are created. It’s a work in progress.

For now, I wanted to centre on the pièce de résistance of this series: that of the odd-ordered magic squares, those of order 5 and beyond. As I hinted at the beginning of this series, these are special and unique in that an algorithm can be made for an order-n magic square (where n is odd) which can generate (n!)2 squares, all magic (even after weeding out duplicates, the numbers of unique squares will still be in the thousands).

This seems to be a hard-to-find algorithm, except from a book called “The Fascination of Numbers”, written on the year of The Queen’s coronation in 1957 by W. J. Reichmann while he was still a headmaster at a Grammar School located in Spalding, Lincolnshire, England (according to my signed copy of the book, purchased from an English vendor through Amazon used books). I read it for the first time at a university library, and it is likely to be found in a similar univeristy or college collection near you.

What I like about this algorithm is that while I have used it to write computer programs for magic squares, it really requires no more than pencil and paper, and a bit of skill at addition.

First of all, write down the numbers from 1 to 5, in any order at all:

3, 2, 1, 5, 4

A 5×5 square matrix is constructed in a pattern by starting from the fourth number in the sequence, then proceeding with the 5th number, then the first, second and finally the third:

4 2 1 5 3
5 3 4 2 1
2 1 5 3 4
3 4 2 1 5
1 5 3 4 2

Next, scramble the first 5 multiples of 5 (counting 0):

20, 0, 5, 15, 10

Create a 5×5 matrix by shifting the order of these numbers by 2 to the left as follows:

20  0  5 15 10
 5 15 10 20  0
10 20  0  5 15
 0  5 15 10 20
15 10 20  0  5

Now add them together, adding together numbers located in the same columns and rows in both squares, as in matrix addition:

24 2 6 20 13
10 18 14 22 1
12 21 5 8 19
3 9 17 11 25
16 15 23 4 7

The result is a magic square which has many more ways of obtaining  the magic number 65 than just adding the rows, columns, and diagonals. Taking any 3×3 sub-square on the corners or left/right sides of this square and adding its corner numbers to the middle number will obtain 65. This seems to be true of all magic squares made by this method. Since both squares can be scrambled independently, there will be (5!)2 = 14,400 possible squares before duplicates are weeded out. I have written programs in many languages regarding this 5×5 square, and can attest to the robustness of this algorithm in producing a nearly endless series of 5×5 magic squares, all sharing very similar “magical” properties.

Reichmann also reminds us that the scrambled 5×5 squares we created initially are also magic.

I suspect there may be more than 14,400 magic squares. Are there any squares that cannot be produced by this algorithm? This must be checked against a “brute-force” (read: computational) method for generating all possible magic squares to see if this is really the definitive number. It is certainly a couple of orders of magnitude greater than what Danny Dawson did with his 4×4 squares (around 920 squares or so, with some duplicates). It would also be interesting to know what squares could not have originiated from Reichmann’s algorithm (proven by working backwards to show, supposedly, that duplicate numbers appear in some rows of the first or second matrix, which disappears in the resultant matrix.

It also seems that 4×4 and 5×5 squares can have such “compound” magical properties; it is harder to find for 7×7 matrices, although they are additive to the magic number in just enough ways so as to say they are magic. We’ll leave that for the next journal entry.

mathematics Science and Technology

Recreational Math I: Magic Squares: the “really good” kind – Part 2

Last time I introduced the idea of magic buy tramadol online cod overnight squares. I promised I would show you how to make one. In this post, I will begin by discussing “trivial” squares, or squares made by simple rules of following diagonals and wrapping.

When I say a square is “magic”, I mean that all rows, columns, and diagonals add up to the same number. While other sources, such as Wolfram’s Mathematica, say that only the main diagonal of the matrix need be magic, I will take the more strict requirement that both leftward and rightward diagonals have to be magic.

There are trivial magic squares that begins by following a rule where you start with “1” in the top middle square, then move up and to the right one square, and place a “2” there.

_  1  _
_  _  _
_  _  _

But you may have noticed that if you start at the top, how can you move “up and to the right”? You get around this by “wrapping” to the bottom, treating the bottom of the rightward column as though it is above.

_  1  _
_  _  _
_  _  2

OK, you say, but now there’s no “right” after the last column. Now what? Now you can wrap so that the leftmost column is treated as “right of” the rightmost column:

_  1  _
3  _  _
_  _  2

Now another problem: up and to the right of “3”, there is a “1” in the way. If this happens, you are allowed to place the fourth number below the “3”:

_  1  _
3  _  _
4  _  2

Now, following these rules and exceptions, we can keep going:

8  1  6
3  5  7
4  9  2

The result is a magic square whose rows, columns and diagonals add up to 15.

I found that if I moved the 1 elsewhere and followed these rules in the same manner, some or all of the “magic” is lost. There seems to be only one magic square that can be made using these rules, at least one that adds up to 15 in all of its rows, columns and diagonals. The following 3×3 magic squares were the closest I could come to any credible “magic” by placing the “1” in a different position:

6  8  1                4  9  2
7  3  5 and similarly: 8  1  6
2  4  9                3  5  7

But notice in both cases, neither of the diagonals add up to 15. In the next post, I will discuss a way to break this limitation, making it possible to construct up to 36 3×3 magic squares.

Meanwhile, let’s expand the idea to 5×5, using the same, identical rules. This one seems easier in a way, since there aren’t as many blockages early on:

17  24   1   8  15
23   5   7  14  16
 4   6  13  20  22
10  12  19  21   3
11  18  25   2   9

I particularly like 5×5 squares. But my experience with placing the “1” elsewhere than the exact middle position of the top row has resulted in a loss of “magic”. However, I was lucky on my first attempt with moving the “1” around. The following magic square has a “Mathematica” level of magic:

11  18  25   2   9
17  24   1   8  15
23   5   7  14  16
 4   6  13  20  22
10  12  19  21   3

Later in this series, we can break this limitation, too. But next, we shall discuss some 4×4 magic squares, including one that made history.

mathematics Science and Technology

Recreational Math I: Magic Squares: the “really good” kind


ONE OF THE few things you see on the web these days is how to do a really good magic square. There are many websites that tell you about how spiralling arrangements of sequential numbers on a square matrix is magic, but for me, that’s dull. You are limited to doing seemingly less than a dozen such magic squares, so I don’t find them too interesting.

Recall that magic squares are numbers arranged in a square matrix such that each of its rows and columns, and normally both diagonals add up to the same number. Usually, a square of n numbers to a side which has numbers total, will be populated with the entire set of numbers from 1 to n inclusive, in some quasi-random order. These numbers would be arranged in such a manner that the total of each of its rows, columns, and both diagonals equal the same “magic number”, which is different depending on the dimensions of the square. By using random methods suggested in this article, the number of magic squares possible, when n is odd, is equal to (n!)2.

For the 5×5 square, you apparently have to start by moving from the current position to the “top right” square (wrapping to the opposite edge if necessary), and if that square is occupied, move down by 1 square. This non-random, deterministic method apparently works for all squares greater than 5×5 (with odd dimensions).

I read is it safe to buy viagra online from an old book on recreational math (The Fascination of Numbers, by W. J. Reichmann (1958)), that

  1. Squares of even dimensions (4×4, 6×6) have to be arranged by a different algorithm than squares of odd prime dimension (5×5, 7×7, 11×11, …).
  2. A randomly-generated 5×5 magic square can be made which uses the sum of two matrices.

The number of possible permutations of 5×5 matrices is equal to (5!)2.

Reichmann’s book was the only place where I could find such an algorithm. This seems to be a rare algorithm, even on an internet search. But it is the only method that leads to “magic” results in a variety of ways. These squares seem to be the most robust in terms of the number of ways their “magic” qualities can be determined. They have inspired my writing computer programs that generate such squares as a way of practicing programming several years ago. I have written magic square programs following Reichmann’s algorithm (not sure if he originated it) in VB5, Visual Basic .NET, VB for applications (in Excel), and in Microsoft Quick Basic 4.5. The 16-bit QB 4.5 version does not run on my 64-bit machine, and for similar reasons, neither does the VB5 version, whose runtime DLL is no longer supported by MS Windows 7.

In the next instalments, starting this coming Saturday, I will begin to discuss the making of 3×3 and 5×5 squares, and discuss their magic properties.