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Close Relations of Magic Squares

October 1st, 2010 by Math Tricks | 3 Comments | Filed in Magic Squares, Math Tricks

Meet the Magic Square Family!

Like magic squares, there are arrangements of numbers similar to magic squares that have interesting properties.  These arrangements are called heterosquares and anti-magic squares.  Unlike magic squares, these numbers do not form the same product when you sum the rows, columns, and diagonals.  Rather, the sums of the rows, columns, and diagonals are all different from each other.


Heterosquares can be thought of as the black sheep of the magic square family.  The sums of their rows, columns and diagonals are not all the same – they are all different from each other.

An example of a (4 x 4) heterosquare:


Heterosquares exist for all tables equal to and larger than 3 rows  x 3 columns.

Anti-magic Squares

Like heterosquares, the sums of the rows, columns, and diagonals of anti-magic  squares are all different.  However, the sums form a sequence of consecutive 2n+2 numbers – cool beans!
Check out this example:
anti-magic square
Do you see the series?  It is 29, 30, … , 38.

Coming up soon on math tricks – a new magic squares game!  Stay tuned . . .

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