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Magic Squares

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Areas of Science
Time Required
Average (6-10 days)
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A magic square is an arrangement of numbers from 1 to n2 in an n x n matrix. In a magic square each number occurs exactly once such that the sum of the entries of any row, column, or main diagonal is the same. You can make several magic squares and investigate the different properties of the square. Can you make an algorithm for constructing a Magic Square? Can you show that the sum of the entries of any row, column, or main diagonal must be n(n2+1)/2? Are there any other hidden properties of a Magic Square? Show the differences between special instances of the Magic Square, like the Lo Shu, Durer, Ben Franklin, or Sator Magic Squares. Can magic squares be constructed in 3 dimensions? You can also investigate other shapes, like magic circles and stars (Alejandre, 2006; Pickover, 2002). Or test the question, "Is there really no math in Sudoku?" (Hayes, 2006)


  • Hayes, B., 2006. Unwed Numbers: The mathematics of Sudoku, a puzzle that boasts No math required!" American Scientist Online: Vol 94, number 1, page 12. Retrieved 5/1/06.
  • Pickover, C. A., 2002. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press.
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Mathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering. Read more

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MLA Style

Science Buddies Staff. "Magic Squares." Science Buddies, 20 Nov. 2020, https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p036/pure-mathematics/magic-squares. Accessed 28 May 2023.

APA Style

Science Buddies Staff. (2020, November 20). Magic Squares. Retrieved from https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p036/pure-mathematics/magic-squares

Last edit date: 2020-11-20
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