Magic squares have come into prominence in popular culture along with
the advent of such mathematics-based games as Sudoku. A magic square is an
arrangement of integers in a square in such a way that the sum of each
horizontal, vertical, and diagonal row is one constant number, the so-called
magic constant.
Magic squares are solved by taking an empty square and inserting the
correct numbers throughout until each line adds up to the magic constant. There
can be both even and odd numbered magic
squares.
Solve an even order
square. Fill out the square starting from left to right. The simplest even
square is a 4 by 4 square, which would contain 16 boxes for the integers. The
magic constant for a 4 by 4 square is 34. Begin with the number 1 in the top
left, and continue until you end with 16 in the bottom right box.
Rearrange all the numbers on one diagonal line. In the
4 by 4 square, begin with the line beginning at 1 and ending at 16. Replace them
with one another, so the entiresequence now begins
with 16 and ends with 1. Then switch the 6 and 11, so that your diagonal line
now reads 16, 11, 6, 1.
o
Rearrange the numbers on the second diagonal line. This line begins with
4 and ends with 13. Switch these numbers and then switch the numbers 10 and 7.
The line should now read, from top right to bottom left, 13, 10, 7, 4.
o
Calculate the sum of each line. Every sum should now be 34. Continue this
method with larger and larger Magic Squares.
2. 2
Solve an odd order square. Start in the top
middle square. The simplest odd square is 3 by 3, which contains 9 boxes for
the integers. The magic constant for this square is 15. Place a 1 in the top
middle box.
o
Move one square up and one square to the right to place your 2. This box
is off thegrid, so move to the
bottom of the grid, as though it wraps around. Place a 2 in the bottom right
hand corner.
o
Move one square up one to the right to place your 3. Again, this is off
the grid. Move as though the grid wraps around and place a 3 in the middle left
hand box.
o
Move up one square and one to the right to place your 4. This box is occupied by the
1, so simply place the 4 in the box underneath the 3. The 4 should now occupy
the bottom left hand corner.
o
Place the 5 in the box one space up and one space the right. The 5 is
now in the middle square.
o
Place the 6 in the box one space up and one space the right. The 6 is
now in the top right hand square.
o
Move up one square and one to the right. Again, this square is off the
grid. Wrapping around, the 7 should go in the bottom left hand corner, which is
occupied by the 4. Simply place the 7 in the box directly below the 6.
o
Move up one square and to the right. Wrapping around the grid, this
space would be the top left hand box. Place the 8 here.
o
Place the 9 in the remaining box, the middle bottom square.
3. 3
Try to use variations of these steps to discover your own solution
methods.
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