A 3 x 3 magic Square dating back to about 2850 B.C. It is attributed to Fuh-Hi, the mythical founder of Chinese civilization, who lived from 2858- 2738 B.C.

The odd numbers are expressed by white dots, i.e. yang symbols, the emblem of heaven.

The even numbers are represented by black dots, the yin symbol, the emblem of earth.

This is the oldest known example of a magic square. It is also the only possible arrangement of the first 9 numbers into a magic square (not counting the 3 rotations and 4 reflections).

12 Magic Circles - 6 Magic Squares

This array contains six 4 x 4 pan-diagonal magic squares which each sum to 194 in 52
different ways.

(4 rows, 4 columns, 2 main diagonals, 6 broken diagonal pairs, corners of 4 - 3 x 3
squares, corners of 16 - 4 x 4 squares including wrap-around, and 16 - 2 x 2 squares
including wrap-around.)

The twelve circles of 16 numbers each sum to 776 ( 4 times 194).

**A** 4 x 4 magic square. All rows, columns, & the 2 main
diagonals sum to

340. Also** **7 sets of 2 x 2 square arrays also sum to this number.

**B** 4 x 4 magic square as A.

**C** 5 x 5 magic square. All rows, columns, & the 2 main diagonals =
425.

**D** 4 x 4 magic square as A. ( 2 of the 7 sets are different then in A
& B).

**E** 4 x 4 magic square as D.

**F** 9 x 9 magic squares. All rows, columns, & the 2 main diagonals =
765.

**G** 5 x 5 magic square as C.

**H** 9 x 9 magic square as F.

**I** 7 x 7 magic square. All rows, columns, & the 2 main diagonals =
595.

**J** 13 x 13 magic square. All rows, columns, & the 2 main diagonals =
1105.

**K** 11 x 11 magic square. All rows, columns, & the 2 main diagonals =
935.

**L** 7 x 7 magic square as I.

**M** 3 x 3 semi-magic square. All rows, columns, but only 1 main diagonal =
255.

All 169 cells in this array are included in at least two different magic squares. The cell containing the number 85 appears in six different magic squares. It is also in the correct diagonal of the 3 x 3 semimagic square!

This array was invented by L.S.Frierson and appears in *Magic Squares & Cubes*
by W.S. Andrews, publ. in 1917.

Spell out the numbers in the first magic square. Then count the letters in these number
words. The integers make a second magic square. This second square contains the
consecutive digits from 3 to 11. This first square is referred to as an 'alphamagic'
square. It was invented by Lee Sallows who made a thorough investigation of this type of
square and reported the results in *Abacus (1986 & 1987). *It has since appeared
in many publications.

**Sign of the Beast**

Add 100 to each cell of the above two magic squares. Now add the corresponding cells
together to make a new magic square. The constant of this new square is 666.

This novelty magic square is known as the IXOHOXI magic square. It is magic in all four of the above orientations. It is pan-diagonal so 4 rows, 4 columns, 2 main diagonals, 6 complementary diagonal pairs and 16 2 x 2 squares all sum to 19998.

**Check this out with a mirror!** All numbers in the reflection will read
correct because both the one and the eight are symmetric about both the horizontal and the
vertical axis. Note also that the name IXOHOXI has the same characteristics.

**Upside-down magic square**

The following digits are correct if rotated 180 degrees

i.e. 0, 1, 6, 8, 9 when rotated 180 degrees becomes

6, 8, 9, 1, 0.This magic square is still magic when rotated 180 degrees

However, if these digits are simply turned upside down the 6 becomes a
backward 9 and the 9 a backward 6.

If you turn the square upside down, then reverse the 6’s and 9’s so they
read correctly, you end up with different numbers, but the square is still magic!

Notice that corners of any 2x2, 3x3 or 4x4 squares also sum to 264, as well as many other combinations.

1 | 143 | 142 |
4 | 5 | 139 | 138 | 8 | 9 | 135 | 134 | 12 |

13 | 23 | 121 | 120 | 119 | 27 | 29 | 31 | 113 | 112 | 30 | 132 |

131 | 117 | 41 | 103 | 102 | 44 | 45 | 99 | 98 | 48 | 28 | 14 |

130 | 105 | 96 | 55 | 89 | 88 | 59 | 84 | 60 | 49 | 40 | 15 |

129 | 39 | 95 | 87 | 65 | 79 | 78 | 68 | 58 | 50 | 106 | 16 |

128 | 107 | 51 | 62 | 76 | 70 | 71 | 73 | 83 | 94 | 38 | 17 |

18 | 37 | 93 | 82 | 72 | 74 | 75 | 69 | 63 | 52 | 108 | 127 |

19 | 36 | 53 | 64 | 77 | 67 | 66 | 80 | 81 | 92 | 109 | 126 |

125 | 35 | 54 | 85 | 56 | 57 | 86 | 61 | 90 | 91 | 110 | 20 |

21 | 111 | 97 | 42 | 43 | 101 | 100 | 46 | 47 | 104 | 34 | 124 |

22 | 115 | 24 | 25 | 26 | 118 | 116 | 114 | 32 | 33 | 122 | 123 |

133 | 2 | 3 | 141 | 140 | 6 | 7 | 137 | 136 | 10 | 11 | 144 |

All these squares are magic in all rows, all columns, and 2 main diagonals.

Magic sums: Order-4, 290; Order-6, 435; Order-8, 580; Order-10, 725; Order-12, 870.

The order-12 is a pure magic square, i.e. it contains the consecutive integers 1 to 144
(12^{2}).

Add |
The normal order 3 magic square. This is the only basic
solution for the order-3. The constant is 15. |

Subtract |
Constructed by interchanging the contents of diagonal
opposite corners. Now, if you add the two outside numbers and subtract the center one from
the sum, you get the constant 5. |

Multiply |
Constructed in the same sequence as the normal order 3 m.s.
Start each sequence of 3 integers by doubling the value of the previous sequence start
number. The 2nd number in the sequence is 3 times the 1st number. The 3rd number in the
sequence is 3 times the 2nd number.. Multiply the 3 numbers in each of the eight lines to
obtain the constant 216. |

Divide |
Construct the same as the multiply m.s., then interchange
diagonal opposite corners. Now, by multiplying the outside numbers of each line, and
dividing by the middle number, the constant 6 is obtained. |

Geometric |
Raise the number in each cell by the power indicated. The
product of the 3 numbers in each row, column, or diagonal will be the magic constant, in
this case 14,348,907. Or, for a short cut, raise the number to the sum of
the powers in each cell of the line. NOTE that the exponents are arranged the same
as in the normal magic square. This works for any base number. |

Numbers 1 to 30 arranged so that the corners of each of the nine
hexagons sums to 93.

Two of many solutions.

"**A**" shows the unique solution for this arrangement of the
integers 1 to 19. The 6 lines of 3 numbers; 6 lines of 4 numbers; and 3 lines of 5 numbers
each sum to 38.

No other solution *for any order hexagon *is possible !!

"**B**" is formed by picking an arbitrary number, in this case
35, and subtracting each number in "A" from it. This arrangement has the
characteristic that each of the 6 lines of 3 integers sum the same, in this case 67. Each
of the 6 lines of 4 integers sum to the same value, in this case 102. And each of the 3
lines of 5 integers sum the same, in this example, 137.

Martin Gardner*, Sixth Book of Mathematical Games pp 22-23 *credits C. W.
Adams

John Hendricks. *A magic Square Course*, page 7, credits H.Lulli

The design is seen in many other books as well, with no credit given.

Addendum: Sept. 12, 2002
Jerry Slocum mailed me a copy of an advertisement (?)
dated 1896, crediting W. Radcliffe, Isle of Man, U.K. with this discovery Jerry Slocum has a large collection of mechanical puzzles. See pictures here of two puzzles based on this magic hexagon as well as puzzles based on magic stars. (Click on this picture for an enlarged view). |

**An Order-9
pandiagonal Magic Square**

The general belief among magic square enthusiasts has been that it is impossible to construct a pandiagonal order-9 magic square.

However, in 1996 Mr. Gakuho Abe discovered a whole series of such squares.

See Dr. Alan Grogono's site at Magic Squares by "Grog" for more information.

This square is composed of the consecutive series of numbers from 1 to 81 and as is usual with pure magic squares, all rows, columns, and the two main diagonals sum to the constant 369.

Being pan-diagonal, the broken diagonals also sum to the magic constant. and the square can be transposed to another by moving any column or row to the opposite side.

**Please send me Feedback about my Web site!**

Harvey Heinz harveyheinz@shaw.ca

Last updated January 28, 2009

Copyright © 1998,2002 by Harvey D. Heinz