# Construction, classification and census of magic squares of order five

by Albert L. Candy

Publisher: A. L. Candy in Lincoln, Neb

Written in English

## Subjects:

• Magic squares.

## Edition Notes

Classifications The Physical Object Statement by Albert L. Candy. LC Classifications QA165 .C25 Pagination v, 141 p. Number of Pages 141 Open Library OL6381738M LC Control Number 39001398 OCLC/WorldCa 7840952

The constant values \$ M \$ of the sums of the magic squares have a minimum value (for non-zero integer positive values). \$\$ M = n (n ^ 2 + 1) / 2 \$\$ For a size 3x3, the minimum constant is 15, for 4x4 it is 34, for 5x5 it is 65, 6x6 it is , then , A 6 6 magic square is then the first to have order 4n + 2. The following is a 6x6 magic square. We will examine the way this particular 6 6 magic square was generated. The are other methods, but as we will see they are not as straightforward as the methods we showed for odd ordered magic squares and doubly-even ordered magic squares. Murphy (Random House, , \$, ISBN ). Although this book is intended for younger readers, it is a great way to introduce the magic square. It tells of Ben Franklin’s many inventions including the story of how he created the magic square. To order, call Visit this site to play magic squares games. Magic Squares of Order 4n Here we will generalize the method used to generate fourth-order magic squares to generate squares of order 4n. That is, squares for which the number of cells on a side is a multiple of 4. We can use almost the same process as we used to generate a fourth-order magic square to create any 4n 4n magic square.

A magic square is a square array of numbers with the property that the sum of the numbers in each row, column and diagonal is the same, known as the “magic sum”. The ‘order’ is the number of rows and columns, so a magic square of order 4 means it has 4 rows and 4 columns. If N is the order, then N x N different numbers are used to.   Universal Magic and Bimagic Squares of Orders 3 to 32 Using 2 and 5. This work brings, magic squares of order 3 to 32 just with two digits in such a way that the magic squares [ ] June 9, 0 Universal Magic and Bimagic Squares of Orders 3 to 32 Using 1 and 8. The basis for construction is the concept of base lines. For the purposes of this discussion a base line for the order-4 magic square is a 4-bit code containing equal numbers of 0's and 1's. The Order-4 Base Square Generator contains two base lines, one for the rows and the other for the columns. These are combined to make a base square. Media in category "Associative magic squares of order 4" The following 48 files are in this category, out of 48 total. Associative magic square 4x4 × ; 17 KB.

## Construction, classification and census of magic squares of order five by Albert L. Candy Download PDF EPUB FB2

Get this from a library. Construction, classification and census of magic squares of order five. [Albert L Candy]. For order 5 squares, these three methods give a complete census of the number of magic squares that can be constructed by the method of superposition. Neglecting the rotation and reflections, the total number of magic squares of order 5 produced by the superposition method is +.

4 The method for even magic squares divisib le by 4 (order 4, 8, 12 etc.) 5 The method for even magic squares not divisible by 4 - “The Strachey Method”.

(Order 6, 10, 14 etc.). - Order-5 pandiagonal magic square showing parity pattern - An unorthodox use of wrap-around 1 4 2. 65 64 8 3. square. 21 35 57 12 76 52 2 59 45 19 48 4 16 62 61 38 24 33 49 11 34 63 23 56 25 54 68 13 30 18 32 58 21 47 70 66 65 7 51 36 77).

Figure Natural magic square of order 5 with constant Lozenge method. Construction of a 4 × 4 magic square using distinct positiv e cubed integers, or proving. This volume explores the science of magic squares and their history and construction from Ancient Times to ADfollowing the discovery of fragments of the earliest Arabic writing on the subject.

It provides a historical outline and English translations of its sources and study. 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. 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. 2 2. Basic facts and definitions A primitive magic square (referred to as a magic square in what follows) of order n is a square consisting of the n2 distinct numbers 1, 2, 3,n2 in n2 subsquares such that the sum of each row, column and main diagonals adds up to the same total, n(n2 + 1)/2.

A double even order magic square is one whose order is divisible by 4. "THE CONSTRUCTION OF MAGIC SQUARES AND RECTANGLES BY THE METHOD OF "COMPLEMENTARY DIFFERENCES."" is an article from The Monist, Volume View more articles from The Monist.

Magic squares of single-even order (n=6, 10, 14, ) are known to be difficult to construct. One reason e.g. can be recognized, if you divide the square in its four quadrants. This time, these quadrants are of odd order, so that it is impossible to fill them in a symmetrical order.

In the construction of my 16×16 panmagic square I kept the sums of pairs restricted to two in both rows and columns, as in the second 12×12 square above.

The result is a square containing 30 panmagic (sub)squares, and 33 semi-magic 4×4 units. Enjoy the beautiful pattern in the positioning of the numbers, going from 1 to 5. STEPS FOR CONSTRUCTION Doubly-even magic squares Construction of magic square using basic Latin square is expressed in the following steps: Step First arrange the consecutive numbers 1 to n2 or (a 11 to a nn) in basic Latin square format.

Find T= > n2 1 @. Step Find the range of 21 aa 11 nn = and select the column. o Generating more magic squares. New but similar magic squares may beconstructed by multiplying each number of the original by a constant multiple.

The sum of the rows will be the multiple of the original sum. For example, multiply each number in Figure 1 by 5. The resulting magic square (Fig. 2) has a column sum of 15 X 5 = Students can.

For the order 7 square, each pan-magic square has 49 x 8 = variations and for the size 11 square there are x 8 = variations.” He also notes that, according to his book on Magic Squares and Cubes, William Andrews describes the construction of panmagic squares of order 5, and predicts that the total number of possible panmagic.

Hence A ∗ B is a magic square and ‖ A ∗ B ‖ = n 3 ‖ A ‖ + m ‖ B ‖. We note that if A and B are magic squares, then each C i j in is also a magic square. Let M S be the set of all magic squares.

Then M S forms a submonoid of M. Lemma 3. For a magic square A of degree n, m A k = m A ⋅ n 2 k − 1 n 2 − 1. Proof. It is clear. A magic square of order n is an arrangement of n 2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.

A magic square contains the integers from 1 to n 2. The constant sum in every row, column and diagonal are called the magic constant or magic sum, magic constant of a normal magic square.

In a magic square, each number needs to use at least two or three series: one for the row, one for the column, and one more if the number is located on a diagonal. Because the number 6 -for example- is present in only one series, a magic square of order n=4 is impossible.

Order n=5, impossible. 7, fourth order normal magic squares and 2, ﬁfth order normal magic squares [4], with higher orders unconﬁrmed [3]. Previous work related to fourth order normal squares has shown that symmetries such as the dihedral group exist [5] and that (under certain conditions) normal magic squares can be categorized into four distinct.

natural nested magic squares of order six. According to [5] the number of super magic squares of order five is sixteen and number of super magic squares of order seven is 20 The number of complete magic squares of order four is 48, and the number of complete magic squares of order eight (cf.

[4]) is a fourth-order magic square with a magic constant of 64 for protection. Construct this square using 7 as the smallest number and 25 as the largest number. Use Pheru's method to construct magic squares in which n equals 5. there is a square of type VI, VII or XI, with that series as one of its diagonals.

Also in each case determine whether a further square can be derived from the given one by ‘tweaking’. • Complete the classification of all magic squares of order 4, stating the total number of squares of each type.

2 the same number of Cells, as each Row, Column or Main Diagonal, i.e. the Order n of the Magic Square. Broken Diagonals also sum the Magic Sum S and are the main characteristic of the Pandiagonal Magic Squares. Main Diagonal – Each one of the two diagonals, called Leading or Left Diagonal and Right Diagonal, constituted with n Cells, which connect the opposite corners of the.

The magic squares of odd order generated by MATLAB show a pattern with increasing elements generally moving diagonally up and to the right. Contents Three Cases Odd Order A New Algorithm Doubly Even Order Singly Even Order Further Reading Three Cases The algorithms used by MATLAB for generating magic squares of order n fall into three cases: odd, n is odd.

doubly-even, n. 4) Start filling the 3 x 3 magic square on the top left with numbers 1 to 9. and top right from 19 to 27, bottom left with 28 to 36 and bottom right with 10 to 5) Now exchange the numbers 8,5,4 from the top left 3 x 3 square to the bottommost left 3 x 3 squares with the numb 32, 31 and vice versa.

6) Your 6 x 6 magic square is ready. In fact, A, C, D are the three basis elements that generate all magic squares of order 3, and T 1 is the unique magic square with magic number 12 up to rotations and reflections.

Theorem 1. Every magic square of order three, up to rotations and reflections, can be written uniquely as either T 1 + iA + jB + kC or T 2 + iA + jB + kD, where i, j. Codes must also be effectively enforced to ensure that buildings and their occupants benefit from advances in seismic provisions in the model codes.

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The science of magic squares witnessed an important development in the Islamic world during the Middle Ages, with a great variety of construction methods being created and ameliorated. The initial step was the translation, in the ninth century, of an anonymous Greek text containing the description of certain highly developed arrangements, no Author: Jacques Sesiano.

(> th), the desired magic square can be achieved. The process of symmetric transformation and adjustment on the pair satisfying T is not required It gives ^ b ij ` satisfying, ¦ j ij i ij b ¦ j ij i ij d [4] ^ b ij `; i j 1, 2.n is the doubly even magic square. STEPS FOR CONSTRUCTION FOR THE IMPROVED TECHNIQUE (For doubly-even magic.

Some modern alternative representations of magic squares: 1. is an order 5 magic square constructed using dowels and metal washers to represent the numbers. Suspending the model from the center demonstrates that it is in balance. an an order 3 square constructed with needlework (cross-stitch).

Numbers: Their Tales, Types, and Treasures. Chapter 7: Placement of Numbers. CONSTRUCTION OF A MAGIC SQUARE OF ORDER 3. A systematic construction of all possible 3 × 3 magic squares would begin by considering the matrix of letters representing the numbers 1 to 9 shown in figure Here the sums of the rows, columns, and diagonals are denoted by r j, c j, and d j, respectively.

Santa Clarita (/ ˌ s æ n t ə k l ə ˈ r iː t ə /) is the third-largest city in Los Angeles County, and the 21st-largest in the state of California. As of the census it had a population ofand in the population was estimated to beThe city has gained much of this rapid population increase by annexing numerous unincorporated areas.

It is located about 30 miles. Updated population controls are introduced annually with the release of January data. Dash indicates no data or data that do not meet publication criteria. Magic Squares Inder J. Taneja, Block-Wise Equal Sums Pandiagonal Magic Squares of Order 4k, Zenodo, Janupp.Inder J.

Taneja, Magic Rectangles in Construction of Block-Wise Pandi.