Magic Square and Optimization

A magic square puzzle is a mathematical and recreational challenge involving a grid of distinct numbers (1 to n×n) arranged in such a way that the sum of the numbers in each row, column, and diagonal is equal. This constant sum is referred to as the magic constant or magic sum. The puzzle's complexity lies in arranging the numbers in the square, ensuring that these specific conditions are met. Magic squares have a rich historical and cultural significance, often appearing in ancient texts and artworks, transcending mere entertainment to symbolize mathematical elegance and order.

The M number represents the magic constant or the sum that every row, column, and diagonal of an n×n magic square must equal. For an n×n magic square, the M number is calculated using the formula:

M = n×(n×n+1)/2

In this formula, n represents the order of the magic square (the number of rows or columns). The M number is a fundamental concept in the study of magic squares, serving as a guiding principle for constructing these intricate mathematical arrangements. Understanding and calculating the M number is essential for the precise formulation and solution of magic square puzzles, contributing to the exploration of mathematical patterns and symmetrical designs.

The pursuit of solving magic square puzzles and comprehending the underlying mathematics not only sharpens analytical skills but also fosters an appreciation for the delicate balance between logic and creativity. It stands as a testament to the enduring allure of mathematical mysteries, captivating minds and inspiring further inquiry into the intricate complexities of number theory and spatial reasoning. Here is an example:

8 1 6
3 5 7
4 9 2

M = 3(9+1)/2 = 15        

Let's formulate the problem in OR-Tools:

1. The sum of all elements in each row is M
2. The sum of all elements in each column is M
3. The sum of all elements in main diagonal is M
4. The sum of all elements in minor diagonal is M
5. All numbers should be different

The google OR-Tools package is used for modeling the problem:

Results:

also they can create beuatiful patterns:

Let's do it for larger magic squares:

Here are some specific applications of magic squares in art:

1. Geometric Art:

Artists often incorporate magic square patterns into geometric art. The ordered arrangement of numbers in a magic square grid can serve as a foundational structure for intricate geometric designs. The symmetrical nature of magic squares lends itself well to creating visually captivating and harmonious geometric artworks.

2. Calligraphy and Illuminated Manuscripts:

The grid-like structure of magic squares complements the precise lines of calligraphic scripts, resulting in aesthetically pleasing pages in religious texts and historical manuscripts.

3. Digital Art and Algorithms:

Digital artists and programmers often use algorithms inspired by magic squares to generate computer-generated artworks. By manipulating the numbers and patterns within magic squares, digital artists can create visually stunning animations, fractals, and interactive artworks. Algorithms based on magic square properties can generate intricate and ever-changing visual experiences.

4. Mosaic Art:

Magic square patterns can be translated into mosaic designs. Each number within the magic square can be represented by a different colored tile, and when arranged according to the magic square properties, they create mosaic artworks with a mathematical order. Such mosaics can be found in architectural elements, interior design, and public art installations.

5. Sculptures and Installations:

Sculptors and installation artists incorporate magic squares into three-dimensional artworks. The numbers within the magic square can be represented by physical objects of varying sizes, colors, or materials.

6. Modern and Conceptual Art:

Contemporary artists often use magic squares as a conceptual framework for their artwork. The symbolism of order, balance, and mystery associated with magic squares serves as a metaphorical foundation for exploring themes such as identity, complexity, and the interconnectedness of various elements in life.

7. Cultural and Symbolic Expression:

In cultures where magic squares have historical and cultural significance, artists incorporate magic square motifs into their work to convey deeper meanings. These motifs might be used in traditional crafts, textiles, or paintings, reflecting the cultural heritage and symbolic importance of magic squares in that particular society.

I would like to dedicate this post to the great scientist H?kan Kjellerstrand , honoring his invaluable contributions in the field of OR. www . hakank . org