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Magic Squares of Squares (are PROBABLY impossible) - Numberphile

Numberphile・33 minutes read

Magic squares of squares involve specific numbers arranged in a 3x3 grid where all columns, rows, and diagonals total to 15, dating back to the Lo Shu in 2200 BC. Various attempts at creating these magic squares are discussed, highlighting the complexity and challenges in finding true magic squares of squares.

Insights

  • The earliest magic square, Lo Shu, dates back to 2200 BC and was discovered by Emperor Yu on a turtle's back, showcasing the ancient origins and legends associated with these mathematical constructs.
  • Finding a true magic square of squares is complex, with various attempts like the Parker Square and Bremner Square falling short due to repeated or non-square entries, emphasizing the challenge and precision required in creating these unique grids.

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Recent questions

  • What are magic squares of squares?

    Magic squares of squares are 3x3 grids with specific numbers arranged so that all rows, columns, and diagonals add up to 15.

  • What is the history of magic squares?

    The earliest known magic square is the Lo Shu, dating back to 2200 BC, discovered by Emperor Yu on a turtle's back.

  • What is the Parker Square?

    The Parker Square is a challenging version of a magic square of squares with all entries being square numbers, falling short due to repeated entries and a diagonal sum discrepancy.

  • How do magic squares scale up?

    Scaling up magic squares involves multiplying all elements by a factor, maintaining the same relationships and structure despite numerical changes.

  • What is the significance of the Parker surface?

    The Parker surface contains 368 known rational or elliptic curves, potentially aiding in the search for magic squares of squares, with uncertainty about the existence of a 3x3 magic square on the surface.

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Summary

00:00

"Challenges in Creating Magic Squares of Squares"

  • Magic squares of squares involve a 3x3 grid with specific numbers arranged in a way that all columns, rows, and diagonals add up to 15.
  • The earliest known magic square is the Lo Shu, dating back to 2200 BC, with a legend of its discovery by Emperor Yu on a turtle's back.
  • Imperfect magic squares allow repeated numbers, making it easier to create but less authentic.
  • A more challenging version involves the squares of numbers in a 3x3 grid, exemplified by the Parker Square with all entries being square numbers.
  • The Parker Square, while close to a magic square, falls short due to repeated entries and a diagonal not adding up correctly.
  • The Bremner Square is another attempt at a magic square of squares but fails due to two entries not being square numbers.
  • Sallows' semi-magic square improves on the Parker Square by having distinct entries but still has a diagonal sum discrepancy.
  • The complexity of finding a true magic square of squares is highlighted through a system of equations representing the conditions required for such a square.
  • The difficulty in finding a magic square of squares is emphasized, with geometric considerations suggesting it may not exist.
  • Analogies are drawn between the challenge of finding a magic square of squares and the familiarity of Pythagorean triangles, showcasing how seemingly different solutions can have underlying similarities.

10:52

Scaling Magic Squares: Multiplying and Constraints

  • Scaling up triangles and magic squares involves multiplying all elements by a factor, maintaining the same relationships.
  • Multiplying Lo Shu magic square elements by 2 results in a new magic square where all numbers are doubled.
  • Scaling magic squares by a factor retains the same structure and relationships, despite numerical changes.
  • Multiplying coordinates in a magic square of squares by 7 results in each entry being multiplied by 49.
  • Solutions to magic square problems can be scaled up or down, even with fractions, maintaining the magic square properties.
  • Equations for magic squares operate in an 8-dimensional space, with constraints reducing the dimensions.
  • Constraints in the 8-dimensional space lead to a 6-dimensional surface, with each constraint reducing the degrees of freedom.
  • The Parker surface represents a 2-dimensional surface where each point encodes a magic square of squares.
  • The Parker surface contains points with integer coordinates, but finding distinct integer solutions is challenging.
  • Diophantine geometry tackles problems like finding integer solutions to polynomial equations, as seen in magic squares.

22:07

Falting's Theorem: Curves on Surfaces Explained

  • The theorem from the early 80s, proved by Faltings, states that surfaces have finitely many rational points.
  • The Mordell conjecture, now Falting's theorem, was a significant achievement in mathematics.
  • Different types of curves on surfaces have varying numbers of points, with some having only finitely many.
  • The Parker surface has a limited number of specific curves, discovered by the speaker and co-authors.
  • The surface can be divided into curves, potentially aiding in finding magic squares of squares.
  • The Parker surface contains 368 known rational or elliptic curves.
  • The Lang-Vojta conjecture suggests there are only finitely many points with integer coordinates outside rational and elliptic curves.
  • The speaker expresses uncertainty about the existence of a 3x3 magic square of squares on the Parker surface.
  • Larger grids, like a 4x4 magic square of squares, are possible and have historical significance.
  • As dimensions increase, the geometry of spaces becomes more positively curved, potentially allowing for more rational and elliptic curves.

33:29

Matt Parker Reacts to Numberphile2 Video

  • Special Matt Parker reacts video on Numberphile2
  • Link provided on screen and in video description
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