Magic Squares, Shapes and Solids

Nicasio Velez holds his senior thesis on magic squares. – Courtesy of Nicasio Velez

Have you ever wondered how Benjamin Franklin and an ancient Chinese turtle can be related? The answer: magic.  

For centuries, mathematicians and non-mathematicians alike have been studying this puzzle referred to as a Magic Square. A puzzle—not dissimilar to Sudoku—that is constantly altered, expanded and redefined to promote more research and ultimately entwine Franklin and turtle.

Now I know many think math is magic: professors pulling variables and symbols out of seemingly nowhere just to make them all disappear again. But, Magic Squares obtain their name from the history and ability to captivate many minds rather than anything Houdini related.

Magic Squares have varying origins depending on the cultures, but the earliest origin dates back to 23 BCE when a storm struck Ancient China. The people of China, desperate to calm the storm, attempted to sacrifice a turtle to the river god Lo.  After many failed attempts, the turtle reemerged from the water with an inscription: the solution to the Magic Square.

The Magic Square is simply a grid of slots with an equal number of rows and columns. The “order” of the square is deemed from how many rows (or columns) the puzzle has. This first ever Magic Square (often referred to as the Lo Shu Square) was a 3rd order, or 3×3 Magic Square. Each slot in the Magic Square is to be filled with consecutive integers (1,2,3,etc.) with the aim of each row, column and diagonal adding to be the same number.  

Many have proved that this turtle’s solution is the only solution to this order of Magic Squares. In fact, try it yourself; if you find multiple solutions, try rotating the square or flipping the numbers around the middle. Because the pattern of numbers stays the same when you rotate or flip the square, this is the same solution. Now try adding a row and a column and see if you can solve that.

There are exactly 880 solutions to the 4×4 Magic Square and so many with the 5×5 that many debate the precise number of solutions.

If you feel comfortable with these, try a 16×16 Magic Square. This is what Benjamin Franklin studied and later deemed “the most magically magical of any magic square ever made by any magician” in a letter to Peter Collinson.  

His solution to this order has countless patterns beyond the desired row and column sums. There are even solutions that contains Magic Squares inside of Magic Squares inside of Magic Squares. I encourage you to play around, if squares aren’t enough try researching Magic Cubes or Magic Polyhedra. Now there’s a challenge.

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