Grogono Photo Pages
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Make Your Own: Any Prime   3x3   4x4   5x5 

 Make Your Own Magic Square: 3x3

The magic square below is the sum
of the patterns that these numbers
make. Use them or enter new ones
and then click
ArrowRight ArrowDown
To understand the structure, move the
mouse over the blue arrows and wait.
Look at the pattern that your selection
produces in the square below
The Bottom Right number
(under the blue down arrow)
above is the first number in
the sequence for the square.
By convention it is "1", but
any number can be used.

If you change the bottom row
above and try using "3,2,1,0"
instead, do you get the same
result as "2,1,0,1"?
You can also try experimenting
by writing in your own numbers
in the red squares. To see the
totals, press
The numbers at the top and
the left are the sums for the
diagonal rows - including the
broken diagonals.


The numbers in the Red Squares form the 3x3 magic Square. The numbers beside the Red Squares show the totals for each row. The horizontal and vertical totals are to the right and below in green squares. The other, blue, squares show the diagonal totals - including all of the "broken diagonals". You can make your own Magic Square in two ways. Try both methods:

  1. Enter your own numbers into the Red Squares and then click on "Add Rows". You can experiment with any numbers using any strategy.

  2. Put numbers in the top set of squares and click on "Make Square". If you try this method try any numbers you like. But, to get a conventional square use:

How does this Work?:

Positioning your Mouse over a Blue Arrow distributes the pattern of components in the Magic Square. This reveals the underlying structure of a 3x3 Magic Square. Actually, all 3x3 Magic Squares have an identical structure. And, if the same numbers are used, e.g., 1 to 9, the same square always results; it may be reflected, rotated, or both, but it is always the same square.

In the 3x3 square, it is impossible to make all of the diagonals "magic". The Main Diagonals are "Magic" when you put the middle value (the "3" and the "1") in the center location in their sequences in the top array. If you put these "middle" numbers in other positions, then one of the broken diagonals becomes magic instead.

Make Your Own: Any Prime   3x3   4x4   5x5 


Copyright © Mar 2010 Magic Squares
Mar 6, 2010