A Math Forum Web Unit
Allan Adler's
Multiplying Magic Squares - page 3
Suzanne's Magic Squares ||
Multiplying Magic Squares: Contents || Exploring the Math
 Recall Step One
The first 3x3 square, square A, contains the numbers from 1 to 9, and we can regard it as a pattern for counting out the numbers from 1 to 9.
Step Two
If we were to count out the numbers from 10 to 18 (i.e. the next 9 numbers) in the same pattern, we would get this 3x3 square:
 Check for Understanding:
Verify that this is a magic square.
Subtract 9 from each entry in this new magic square. What do you have?
Another way to say the same thing is that we can get this 3x3 square by
adding 9 (the number of boxes in square A) to each of the entries of A.
Notice that in the big 4x4 grid (the 12x12 magic square broken into 3x3 blocks), this second 3x3 square is found in the bottom row, third position - just where the number 2 is located in square B.
Step Three
Locate the number 3 in square B and put the next 9 numbers (19-27) in the 4x4 grid in that position: that is, add 9 to each of the entries of
to obtain
and put this small magic square in the bottom row, 2nd column of the 4x4 grid.
Another way to say the same thing is that we can get this third 3x3 square by adding 9 (the number of boxes in square A) to each of the entries of A.
Notice that in the 12x12 magic square, this third 3x3 square is found in the bottom row, second column.- the position of the number 3 in square B.
Check for Understanding:
Where would the next newly generated 3x3 magic square be placed?
Where would the last newly generated 3x3 magic square be placed?
Complete the 12x12 Magic Square
After completing the process of generating A*B, try multiplying another two magic squares.
Suggestions:
- Arrange the students in groups of 4.
- Have two of the students in each group find (or generate) a 3x3 magic square while the other two students in each group find (or generate) a 4x4 or 5x5 magic square. Each pair checks to verify that the arrays are magic squares.
- Each group follows the method for finding the 'product'.
- After a magic square 'product' has been generated, groups check the sums of all rows, columns, and diagonals using calculators in pairs. One student reads and makes notations while the other manipulates the calculator. The four students compare their results to verify that they have made a magic square.
Extension
Challenge students to find the 'product' of a 3x3 multiplied by another 3x3 (or other sizes) of magic squares: generate B*A or A*C and C*A, where C is one of the squares derived from A by symmetry.
How to construct a magic square
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