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Contents
Loh-Shu (The scroll of the
river Loh
 |
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).
Addendum: Feb. 23, 2012 Received
from Paul Michelet
The sum of the squares of each line in the Luo Shu square (
i.e. 4922 + 3572 +8162) equals the
sum of the same numbers reversed; both horizontally and vertically. |
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.)
Plus other combinations. See my pandiag.htm
page.
The twelve circles of 16 numbers each sum to 776 ( 4 times 194).
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A
Combination of 13 Magic Squares
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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. |
Alphamagic Number
Square
 |
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. |
Ixohoxi
 |
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.
The source of the square is
unknown.
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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. |
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| 12 x
12 Concentric Magic Square 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 (122).
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|
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 |
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Order 3
Magic Squares - - 5 kinds
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The normal order 3 magic square. This is the only
basic solution for the order-3.
The constant is 15.
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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.
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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.
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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.
|
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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.
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| Magic Hexagons
Numbers 1 to 30 arranged so that the corners of each of the nine
hexagons sums to 93.
These are two of many solutions.
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"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.
Charles
W. Trigg,
The Magic Hexagon,
JRM:16:4:1983-4:234. This complete article is now available
here.
| 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 in 1895. So he beat out Adams!
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). |
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Double Six Magic Square
Buried in the foundation of a house to dispel evil spirits, the Double Six
Magic Square at the Shaanxi History Museum contains six numbers in length
and breadth, the numbers
in vertical, horizontal and diagonal lines add up
to 111, respectively. Thanks to Paul Michelet for
drawing this to my attention.
The 4 x 4 numbers in the inside array also form a magic
square. this one is pandiagonal. Each row, column and tall diagonals sum to 74.
| 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.
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