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## Going Into The Square

Published on Sunday, November 20, 2005 in , ,

Just when you think there's nothing more amazing about magic squares, someone goes and proves you wrong. I've run across some amazing magic squares, including ones that are gigantic, or work with multiplication instead of addition, or can be toured by a chess knight and so on.

Here's an 8 by 8 magic square. See if you can guess what is so unique about it:

`14    56    55    54    53     7     8    131     24    16    45    44    43    23    6463    15    40    26    27    37    50     262    48    29    35    34    32    17     34     47    33    31    30    36    18    615     19    28    38    39    25    46    6059    42    49    20    21    22    41     652     9    10    11    12    58    57    51`

True, it does total 260 horizontally, vertically and diagonally, and uses all the numbers from 1 to 64, but that is hardly unique among magic squares.

Here comes the truly magic part. Let's see what would happen if we removed the top and bottom rows, as well as the leftmost and rightmost columns:
`24    16    45    44    43    2315    40    26    27    37    5048    29    35    34    32    1747    33    31    30    36    1819    28    38    39    25    4642    49    20    21    22    41`

Surprise! We still have a magic square! This 6 by 6 square totals 195 horizontally, vertically and diagonally.

Would you believe we can trim this magic square again, and still get yet another magic square?
`40    26    27    3729    35    34    3233    31    30    3628    38    39    25`

This 4 by 4 magic square totals 130 in all the directions of the previous squares!

These unique constructs are referred to as nested magic squares, and it is possible to create them for any size square starting at 5 by 5.

The above examples were taken directly from the Nested Magic Squares webpage, which teaches methods for creating your own nested magic squares. For those of you who use Matlab, or an Excel-compatible spreadsheet, files are included so you can experiment with these squares yourself.

Just for fun, the author has also included some ginormous magic squares in ascii format, like the 99 by 99 nested magic square (multiple monitors may help in viewing this properly)!

No, there's no practical use for nested magic squares yet (which I consider to be one of their better qualities), but I'll take the experience of wonder wherever I can get it.