Magic Square

1.  Arrange the numbers 1 through 9 in a 3 by 3 array -- a Magic Square -- such that the sums of each row, each column, and the two diagonals are the same.

2.   Is your solution unique? That is, aside from rotation of the square, is there only one way to enter the digits?

3.   Find other 3 by 3 magic squares using distinct entries other than 1 through 9.

a.    Consecutive digits  n  through n +  8 ?

b.    Nine digits in an arithmetic sequence? 

c.    Is it possible to create a magic square with 9 digits NOT in an arithmetic sequence?

For example, either  a.   or   b.  above could lead to a a 3 by 3 magic square where the middle square has 21 entered in it. (Each of the other 8 squares would have an entry other than 21.)

4.   Complete a 4 by 4 magic square with the numbers 1 through 16 for entries?  Is it unique?

5.   Find a 3 X 3 magic square where the operation is multiplication rather than addition and the entries are 9 different numbers.

6.   Can a 3 X 3 magic square be completed with fractions for the entries?

Generating a 5 X 5 Magic Square

A magic square of order n, where n is odd, can be generated by a technique called the Siamese Method.

See  HERE for a demonstration of the generation of the magic square at the right.

1. Begin with 1 in the center position of the top row.

2. Number the squares consecutively to the right one position and up one position. If you are at the edge of the square, loop back to the opposite border.

3. If the position by rule #2 is already filled, move down one row for the next entry and then continue by rule 2.

See if you can generate the 3 X 3 magic square using the Siamese Method. 

See if you and generate the 5 X 5 magic square given above.

Try a 7 X 7 magic square.

There are several good web sites for Magic Square material.   Try a Google Search.

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