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Allan Adler and Suzanne Alejandre:

Why there are no 2x2 magic squares



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Suzanne's Magic Squares || Multiplying Magic Squares: Contents || Exploring the Math
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In explaining composite magic squares, Allan Adler writes:
"... the 12x12 magic square A*B is composite. On the other hand, the 4x4 magic square B is not composite, because the only way it could be the product of two strictly smaller magic squares would be if both were 2x2, and it is easy to see that there are no 2x2 magic squares."
Let's explore two methods of verifying that there are no 2x2 magic squares.

    Remember that in a magic square,

    1. the sums of rows, columns, and diagonals are equal,
      and
    2. the entries are distinct (different numbers).
     

A visual approach

    Let's start with a 1 in the upper lefthand corner, and consider all of the possible 2x2 squares using the numbers 1, 2, 3, and 4.

1 3       1 2       1 4
2 4       4 3       2 3

1 2       1 3       1 4
3 4       4 2       3 2

    A quick comparison of the sums will convince you that these are not magic squares.

    If we start with 1 in the upper righthand corner, there are 6 more possibilities to consider. You can visualize them by imagining the above 6 arrays rotated 90 degrees clockwise. Again, we can quickly see that these next 6 arrays do not qualify as magic squares.

    Another rotation will give us 6 arrays with 1 in the lower righthand corner, and one last rotation will show the possibilities if the 1 is in the lower lefthand corner.

    After considering all 24 possibilites, we can conclude that

there are no 2x2 magic squares.
 

A mathematical approach

Imagine the following magic square:

A B
C D

then [remembering that the sums of rows, columns, and diagonals are equal in a magic square]

A+B = A+C

which [since A can be subtracted from both sides of the equation] means that

B=C

and this contradicts the fact that the entries of a magic square are distinct.

Therefore,

there are no 2x2 magic squares.
 
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