A magic square is a square grid of numbers where the sums of the numbers in each row, each column, and both main diagonals are the same. It is a fascinating mathematical concept that has been studied and explored for centuries. In a magic square, each number is unique and can only appear once. The most common type of magic square is the 3x3 square, where the numbers 1 to 9 are arranged in a grid. However, magic squares can be of different sizes, including 4x4, 5x5, and so on. The term "netdire" refers to the magical properties of a particular magic square.
I was wondering if there were more such rules and where I might find them?
ordinary vector break vector sumdiffs magic squares panmagic squares 1, 0, 1 1, 3 none 1, 0, 2 0, 2 none 2, 1 1, 1, 2, 3, 4 none 2, 1 1, 0, 1, 2, 3 2, 1 1, 0 0, 1, 2 none 2, 1 1, 2 0, 1, 2, 3 none. A generalization of this method uses an ordinary vector that gives the offset for each noncolliding move and a break vector that gives the offset to introduce upon a collision.
The term "netdire" refers to the magical properties of a particular magic square. It is believed to have originated from ancient mystical practices where the arrangement of numbers in a square was thought to hold special powers or significance. The term "netdire" may be specific to a certain culture or region.
Magic Square
A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, . arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number (Kraitchik 1942, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacoby 1981, p. 3; Ball and Coxeter 1987, p. 193), known as the magic constant
If every number in a magic square is subtracted from , another magic square is obtained called the complementary magic square. A square consisting of consecutive numbers starting with 1 is sometimes known as a "normal" magic square.
The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu. A version of the order-4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called Dürer's magic square. Magic squares of order 3 through 8 are shown above.
(Hunter and Madachy 1975).
It is an unsolved problem to determine the number of magic squares of an arbitrary order, but the number of distinct magic squares (excluding those obtained by rotation and reflection) of order , 2, . are 1, 0, 1, 880, 275305224, . (OEIS A006052; Madachy 1979, p. 87). The 880 squares of order four were enumerated by Frénicle de Bessy in 1693, and are illustrated in Berlekamp et al. (1982, pp. 778-783). The number of magic squares was computed by R. Schroeppel in 1973. The number of squares is not known, but Pinn and Wieczerkowski (1998) estimated it to be using Monte Carlo simulation and methods from statistical mechanics. Methods for enumerating magic squares are discussed by Berlekamp et al. (1982) and on the MathPages website.
A square that fails to be magic only because one or both of the main diagonal sums do not equal the magic constant is called a semimagic square. If all diagonals (including those obtained by wrapping around) of a magic square sum to the magic constant, the square is said to be a panmagic square (also called a diabolic square or pandiagonal square). If replacing each number by its square produces another magic square, the square is said to be a bimagic square (or doubly magic square). If a square is magic for , , and , it is called a trimagic square (or trebly magic square). If all pairs of numbers symmetrically opposite the center sum to , the square is said to be an associative magic square.
Squares that are magic under multiplication instead of addition can be constructed and are known as multiplication magic squares. In addition, squares that are magic under both addition and multiplication can be constructed and are known as addition-multiplication magic squares (Hunter and Madachy 1975).
Kraitchik (1942) gives general techniques of constructing even and odd squares of order . For odd, a very straightforward technique known as the Siamese method can be used, as illustrated above (Kraitchik 1942, pp. 148-149). It begins by placing a 1 in the center square of the top row, then incrementally placing subsequent numbers in the square one unit above and to the right. The counting is wrapped around, so that falling off the top returns on the bottom and falling off the right returns on the left. When a square is encountered that is already filled, the next number is instead placed below the previous one and the method continues as before. The method, also called de la Loubere's method, is purported to have been first reported in the West when de la Loubere returned to France after serving as ambassador to Siam.
A generalization of this method uses an "ordinary vector" that gives the offset for each noncolliding move and a "break vector" that gives the offset to introduce upon a collision. The standard Siamese method therefore has ordinary vector (1, and break vector (0, 1). In order for this to produce a magic square, each break move must end up on an unfilled cell. Special classes of magic squares can be constructed by considering the absolute sums , , , and . Call the set of these numbers the sumdiffs (sums and differences). If all sumdiffs are relatively prime to and the square is a magic square, then the square is also a panmagic square. This theory originated with de la Hire. The following table gives the sumdiffs for particular choices of ordinary and break vectors.
| ordinary vector | break vector | sumdiffs | magic squares | panmagic squares |
| (1, ) | (0, 1) | (1, 3) | none | |
| (1, ) | (0, 2) | (0, 2) | none | |
| (2, 1) | (1, ) | (1, 2, 3, 4) | none | |
| (2, 1) | (1, ) | (0, 1, 2, 3) | ||
| (2, 1) | (1, 0) | (0, 1, 2) | none | |
| (2, 1) | (1, 2) | (0, 1, 2, 3) | none |
A second method for generating magic squares of odd order has been discussed by J. H. Conway under the name of the "lozenge" method. As illustrated above, in this method, the odd numbers are built up along diagonal lines in the shape of a diamond in the central part of the square. The even numbers that were missed are then added sequentially along the continuation of the diagonal obtained by wrapping around the square until the wrapped diagonal reaches its initial point. In the above square, the first diagonal therefore fills in 1, 3, 5, 2, 4, the second diagonal fills in 7, 9, 6, 8, 10, and so on.
An elegant method for constructing magic squares of doubly even order is to draw s through each subsquare and fill all squares in sequence. Then replace each entry on a crossed-off diagonal by or, equivalently, reverse the order of the crossed-out entries. Thus in the above example for , the crossed-out numbers are originally 1, 4, . 61, 64, so entry 1 is replaced with 64, 4 with 61, etc.
A very elegant method for constructing magic squares of singly even order with (there is no magic square of order 2) is due to J. H. Conway, who calls it the "LUX" method. Create an array consisting of rows of s, 1 row of Us, and rows of s, all of length . Interchange the middle U with the L above it. Now generate the magic square of order using the Siamese method centered on the array of letters (starting in the center square of the top row), but fill each set of four squares surrounding a letter sequentially according to the order prescribed by the letter. That order is illustrated on the left side of the above figure, and the completed square is illustrated to the right. The "shapes" of the letters L, U, and X naturally suggest the filling order, hence the name of the algorithm.
Variations on magic squares can also be constructed using letters (either in defining the square or as entries in it), such as the alphamagic square and templar magic square.
Various numerological properties have also been associated with magic squares. Pivari associates the squares illustrated above with Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon, respectively. Attractive patterns are obtained by connecting consecutive numbers in each of the squares (with the exception of the Sun magic square).
To create a magic square, there are different methods and algorithms that can be used. One well-known algorithm is the Siamese method, where the numbers are filled in a specific pattern. Another method is the De la Loubère method, which involves a systematic process of filling the square. Magic squares have captivated mathematicians, puzzle enthusiasts, and even artists throughout history. They have been used for recreational purposes, as brain teasers, and as a form of divination. In art, magic squares have been incorporated into paintings and designs to add an element of mystery and intrigue. The study of magic squares has led to various discoveries and mathematical theories. They are connected to number theory, combinatorics, and group theory. Magic squares can also be generalized to higher dimensions, where the sums need to be the same in additional directions. In conclusion, the concept of a magic square is a fascinating mathematical phenomenon. The term "netdire" may refer to the magical properties associated with a particular magic square. They have been studied and explored for centuries, and their patterns and properties continue to capture the imagination of people around the world..
Reviews for "The Role of Magic Square Netdire in Numerology"
1. John - 2 stars
I was really disappointed with the Magic Square Netdire. The product seemed promising, but it didn't live up to its claims. The build quality was poor, and the netdire was flimsy and cheaply made. Additionally, the instructions were unclear and confusing, making it difficult to set up and use. Overall, I would not recommend this product to others.
2. Samantha - 1 star
I found the Magic Square Netdire to be a complete waste of money. The netdire didn't sit flat on the ground, causing the balls to bounce off and creating a frustrating experience. The design was flawed, and the product didn't function as advertised. It was a frustrating and disappointing purchase, and I regret buying it.
3. David - 2 stars
I thought the Magic Square Netdire had the potential to be a fun game, but it fell short of my expectations. The netdire didn't hold up well against repeated use, and the net started to fray after just a few games. The ball also didn't bounce consistently, making the game less enjoyable. I wouldn't recommend this product unless they improve its quality and durability.
4. Jessica - 1 star
The Magic Square Netdire was a complete letdown. The balls kept getting stuck in the netdire, and it was a hassle to untangle them every time. The netdire also didn't stay securely in place, and it would often move or collapse during the game. It was frustrating and not worth the money at all. Save yourself the disappointment and avoid this product.