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Thursday, March 5, 2015

Throwback Fact: Magic Squares

An example of a 4th order Magic Square
Last week we discussed Japanese geometry puzzles. If you missed it, check out Sangaku here. This week, we’ll talk about a puzzle game again: Magic Squares.

These fascinating puzzles date back to around 2200 B. C. E. Traditionally, they are played on a grid consisting of n-rows and n-columns, for a total of n^2 boxes. Each of the boxes are filled with integers so that the sums of the horizontal rows, vertical columns, and diagonals are equal. If each integer from 1-n is used, the square applies for special qualifications. These squares are known as nth order magic squares, and the sum of each row will always be equal to: [n(n^(2)+1)]/2.

While squares where n=3 are most common, we know a lot about 4th order squares. French mathematician Bernard Frénicle de Bessy posthumously published each of the exactly 880 unique 4th order squares after his death in 1693.

Benjamin Franklin
In 1769, Benjamin Franklin invented his own version of the magic square. His is 8 rows by 8 columns, and each row and column has a sum of 260.. There are many other symmetries in the Franklin Magic Squares that perhaps Franklin wasn’t even aware of. The most fascinating part of this version is that we do not know Franklin’s method for constructing the squares. He claimed to a friend in a letter that he could construct the squares are fast as he could write, but we have no modern methods that can crack the squares particularly quickly.

In modern times, the Magic Square has expanded to include more dimensions. Researcher John Hendricks has in fact discovered a perfect magic tesseract (four-dimensional cube) of  the 16th order.

Do these puzzles remind anyone else of Sudoku? Does Sudoku seem easier now that we've talked about Magic Tesseracts?

Leave a comment if you have any questions or comments. Don't forget to check out our giveaway here; you could win  a fabulous Pi Day prize worth over $110!


  1. Some interesting background history on Magic Squares in art, science and culture on this blog:

  2. The kids need to play to enhance their academic achievements as well as develop the sense of joy and satisfaction with playing a musical instrument well. japanese magic method article