## #2 Forces: The Magic Square

In this and the next three issues of Mathamagically Speaking, I will explore four different ways of forcing a number on a spectator. This month I will cover The Magic Square. Specifically we will be covering the Pandiagonal Magic Square.

Traditionally a Magic Square is a square array of numbers (integers specifically) where the columns, rows and two main diagonals all add to the same number.

In this example, the three sided square adds up to 15 in all 8 directions. The "Order" of this magic square is 3 since it is 3x3. The "Magic Constant" in this example is 15.

There are many types of magic squares that add more ways in which the magic constant can be found. Below is an Euler diagram of requirements of some types of 4 × 4 magic squares. Each new layer adds a new type of matching while also including all the preceding matching. Cells of the same color sum to the magic constant.

In a Pandiagonal or PanMagic Square, selecting any four adjacent numbers in the same __row__, __column__, or __any diagonal__ adds to the desired force number. This includes diagonals that wrap off the edge of the square.

**How** **To** **Calculate** **A 4 X 4 Pandiagonal Square**

First each square is identified by an alphabetic label (A thru P).

A |
N |
G |
L |

O |
D |
I |
F |

J |
E |
P |
C |

H |
K |
B |
M |

Next, play with these two variables to arrive at the desired force number... For a 4x4 square, the force number (magic constant) will be (S x 4) + (30 x D)

- S = Smallest number in the square: 7 (example)
- D = Difference between each number: 2 (example)

So in this example (7x4) + (30x2) = 28 + 60 = 88 for the Magic Constant.

Now, fill in the below table, starting with your smallest number and adding the "difference between each number" as you go across.

A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |
L |
M |
N |
O |
P |

7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | 33 | 35 | 37 |

Now fill in the square, placing each number in the corresponding letter cell.

7 |
33 |
19 |
29 |

35 |
13 |
23 |
17 |

25 |
15 |
37 |
11 |

21 |
27 |
9 |
31 |

This square adds to 88 in all directions. Try it!

Reference The Zen of Magic Squares, Circles, And Stars' by Clifford A. Pickover, Ph.D., 2002, pg 68

If we now take this exact square and duplicate it so we have two magic squares over two more magic squares... or an 8x8 grid, we can use this for a force. This is shown below with shading for clarity, but for performing the squares should all look the same.

7 | 33 | 19 | 29 | 7 | 33 | 19 | 29 |

35 | 13 | 23 | 17 | 35 | 13 | 23 | 17 |

25 | 15 | 37 | 11 | 25 | 15 | 37 | 11 |

21 | 27 | 9 | 31 | 21 | 27 | 9 | 31 |

7 | 33 | 19 | 29 | 7 | 33 | 19 | 29 |

35 | 13 | 23 | 17 | 35 | 13 | 23 | 17 |

25 | 15 | 37 | 11 | 25 | 15 | 37 | 11 |

21 | 27 | 9 | 31 | 21 | 27 | 9 | 31 |

A spectator can select ANY of the 64 squares and then select in any available direction 4 squares adjacent to that square either vertically, horizontally or diagonally. The 4 squares will add up to the force number.

How to use the square to force a number

- First think of a routine where you want to force a number. Determine what number(s) fit the routine. If for example you are thinking of a Foursome's Golf Scores, you would want numbers in a reasonable range for golf. If you are forcing a year, you want a number in that range.
- Next determine the numbers S and D (above) that will meet your criteria.
- Fill in the force square. Duplicate it so it is 8x8 and dress it up to match your story line.
- Now have the spectator select one square. For example, they could throw a dart at the 8x8 square.
- Wherever the dart lands they can chose a direction... vertical, horizontal or diagonal.
- They add the selected square and the 3 adjacent squares in the selected direction. This will total the force number

7 33 19 29 7 33 19 29 35 13 **23**17 35 13 23 17 25 15 37 11 25 15 37 11 21 27 9 31 21 27 9 31 7 33 **19**29 7 33 19 29 35 13 23 17 35 **13**23 17 25 15 37 11 25 15 37 11 21 27 9 31 21 27 9 31

There are many routines that can use a Magic Square. What are your favorites? How would you adapt this principle for your own routine?

This article was edited by Karl Rabe

Note from the Editor...

Here is an excel spreadsheet you can use to generate a 4x4 magic square.