The Korean king's magic square: a brilliant algorithm in a k-drama (plus geomagic squares)
Mathologer・23 minutes read
The text discusses various aspects of magic squares, such as their construction methods, historical significance, and the introduction of geometric magic squares. Mathematicians like Euler and Ramanujan have researched magic squares, with King Sejong the Great inspiring a unique method for creating them based on his life. Different techniques, such as Lee Sallows' approach using shapes, are explored for constructing geometric magic squares, showcasing the creativity and complexity involved in these mathematical puzzles.
Insights
- Magic squares exhibit a fascinating property where not only rows and columns but also diagonals add up to the same sum, as seen in the Lo Shu magic square with a total of 15.
- The creation of magic squares, as explored through King Sejong the Great's method, involves intricate analysis of entries to prevent duplication, leading to the discovery of geometric magic squares that can be constructed using various shapes and complex algorithms, showcasing the intersection of mathematics and creativity.
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Recent questions
How do magic squares work?
Magic squares are mathematical puzzles where the numbers in each row, column, and diagonal add up to the same sum. This creates a sense of magic as the numbers align perfectly to form the same total in various ways within the square.
Who discovered the 8x8 magic square?
Srinivasa Ramanujan, a renowned mathematician, discovered an 8x8 magic square with numbers from 1 to 64 and a magic constant of 260. His work in magic squares contributed significantly to the field of mathematics.
What inspired the creation of magic squares?
The Korean drama "Tree with Deep Roots" inspired a method for creating magic squares based on King Sejong the Great's life. The method involves fitting a 3x3 magic square into a lunchbox, revealing a square inside a square pattern.
How are geomagic squares different from regular magic squares?
Geomagic squares introduce a new concept where pieces with areas form a square table of numbers adding up to the same magic number. Converting a magic square into a geomagic square involves replacing numbers with rectangles of corresponding sizes, creating a unique geometric twist.
How are computers used in creating magic squares?
Computers play a crucial role in searching and creating geometric magic squares by piecing together shapes made of different numbers of little squares. This technological advancement has led to the discovery of numerous magic sets and shapes, expanding the possibilities within the realm of magic squares.
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Summary
00:00
"Magic Squares: Geometric Patterns and Mathematical Mysteries"
- The square consists of three rows that combine to form the same square, achievable by cutting a square into three parts in three ways.
- Each column in the square also combines to form the same square, presenting a unique feature.
- The diagonals in the square also combine to form the same square, showcasing geometric magic.
- The Lo Shu magic square, an ancient 3x3 square with numbers from 1 to 9, has rows, columns, and diagonals adding up to 15.
- Mathematicians like Leonhard Euler, Srinivasa Ramanujan, and Pierre de Fermat have researched magic squares.
- Ramanujan discovered an 8x8 magic square with numbers from 1 to 64 and a magic constant of 260.
- The Korean drama "Tree with Deep Roots" inspired a method for creating magic squares, based on King Sejong the Great's life.
- The king's method involves fitting a 3x3 magic square into a lunchbox, revealing a square inside a square pattern.
- The king extends the method to create larger magic squares like 5x5 and 7x7, with all rows, columns, and diagonals summing to the same number.
- The method works for odd-sized magic squares but not for even-sized ones, prompting a quest to understand why it always works, leading to a mathematical proof.
14:33
Creating Geomagic Squares with Lee Sallows' Method
- The process involves analyzing the left and right sides of a chosen entry in a square to avoid duplication.
- Entries in the same row are examined to ensure no duplication occurs after collapsing the square.
- Entries above the middle horizontal line are considered safe from duplication after collapsing.
- Entries below the middle row are analyzed to prevent duplication of specific numbers.
- The 5x5 and 7x7 magic squares in the king's study are constructed differently from Bachet's method.
- Geometric magic squares, or geomagic squares, are introduced as a new concept.
- Geomagic squares involve pieces with areas that form a square table of numbers adding up to the same magic number.
- Converting a magic square into a geomagic square involves replacing numbers with rectangles of corresponding sizes.
- Lee Sallows' method for creating nontrivial geometric magic squares involves using shapes like semicircles, rectangles, and squares.
- Computers are used for searching and creating geometric magic squares by piecing together shapes made of different numbers of little squares.
29:02
"Discovery of BI-magic squares with 12 shapes"
- 1411 sets of pieces form magic squares with a 3x5 rectangle as the magic constant shape, with almost 30000 different magic sets for a 4x4 square with one corner missing, leading to the discovery of a BI-magic square with 12 different magic constant shapes.
