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| image: commons.wikimedia.org 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.
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| image: en.wikipedia.org 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.
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| image: commons.wikimedia.org Tesseract |
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!
Some interesting background history on Magic Squares in art, science and culture on this blog: www.glennwestmore.com.au
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