Contents
Order6 Characteristics 
Introduction and some facts about
Order6. 
First Solution Solvers 
A short history of the early solvers of magic
hexagrams. 
Complements 
Each solution has a complement, just
like magic squares. 
Adjacent Complement Pairs 
A table showing the complement pairs
that are adjacent in the list of basic solutions. i.e. have
consecutive index numbers. 
Twenty Sets of Four 
Mr. Suzuki's groups, diagram of
transformations & list. 
Supermagic Stars 
Six is the only order that can have
the sum of peaks also magic. 
Cell Pairs equaling 13 
6 pairs of cells define a way to put
the 80 solutions into three groups. 
Related pages on this
site 
Use the browser 'back' button to return here if you
wish. 
Definitions 
Review magic star definitions and
general features. Use Back button to return. 
A Tribute to H. E. Dudeney 
Applying all of Dudeney's findings to
my solution list. 
All Order6 Basic Solutions 
All 80 basic solutions in index
order, including complement #, how they fit in Mutsumi Suzuki's 20
sets of 4, H.E. Dudeney's classification and several other features. 
Order6
Characteristics

Order6 is the lowest order
pure magic star. It is also the only order (at least to order12) that
has more basic solutions then a higher order. The only exception is
order10a, which has more solutions then any of the four
order11patterns.

There are a total of 80 basic solutions
consisting of 40 complement pairs.
The 80 solutions may be subdivided into 20 sets of 4 solutions (Mutsumi
Suzuki). They may also be classified into three groups on the basis
of which two or three pairs of cells sum to 13 (Dudeney).

Each solution contains the consecutive
series of integers from 1 to 12. The sum of this series is 78, there are
6 lines with each cell appearing in 2 lines, so the magic constant (S)
is (2 x 78)/6 = 26.

It is also the only order star where there
are solutions that have all the (points)
summing to the magic constant. There are six such solutions, and by
complementing, there are also six solutions with the valleys constant.

There can be no solutions with the points
consisting of consecutive numbers because six consecutive numbers must
always sum odd and all solutions have the points summing even. There are
no solutions with the points all even or all odd.

Solution # 16, shown above, is the first of
six supermagic stars, because the points also sum to 26. Solution # 79
is the last of six solutions where the valleys sum to 26.

Order6 is the only magic star that does not
have at least one continuous pattern. This pattern consists of two
superimposed triangles. (2 and 3 are the factors of 6). The sum of the
large triangle points are always equal.

Each pattern contains 3 diamonds such as
a, c, h, k whose points always sum to 26.
Each of the three opposite pairs of small triangles always sum to the
same value.

There are 21 solutions where a + c
= 13. In each case, and only these 21 cases, b + d, f +
l and h + k also sum to 13. In addition, e + i
and g + j also sum to 13, but these two pairs also sum to 13 in
seven other solutions. (Cell pairs =
13).
First Solution
Solvers
In 1917, W. S. Andrews showed a magic hexagram using numbers 1 to 13,
with no 10. [1]
In 1926,H. E. Dudeney published 74 solutions for the magic hexagram
using the consecutive numbers 1 to 12. He erroneously thought that was all
there were. [2]
He discovered many interesting features of this smallest of the normal
magic stars. See my Dudeney
page.
Martin Gardner reported in an Addendum to his 1965 article that E. J.
Ulrlich, of Enid, Okla, USA, and A. Domergue of Paris, France, both
discovered there are actually 80 solutions for the basic magic hexagram.
However, he did not give dates for these discoveries.
[3] I suggest that they were
probably made between 1965 and 1975, because in the original column,
Gardner mentions that there are 74 basic solutions and first
mentions the 80 in the 1975 book!
Recently (2005), while reviewing J. R. Hendricks collection of notes, I
found a reference to an early finding of all 80 solutions.
[4]
John Hendricks recorded
“Mr. (L. M.) Leeds wrote that he found all 80
basic magic hexagrams in 1932. More recently, he computerized them and
found no more.”
I expect that due to the short period after Dudeney’s publication, Mr.
Leeds was probably the first to find all 80 solutions! (Laurance M. Leeds
was a graduate of the School of Engineering 1934, Rutgers University in
New Jersey, USA. He was inducted into their Hall of Fame posthumously in
2006 for his work on early television and radio.)
In his unpublished book [4], John
Hendricks included a computer program he had written to find the 80
solutions.
As a point of interest, about 4 years later (1995), I also wrote a basic
program to find these 80 solutions. At that time I had still not located
(seen) any list of the 80 solutions (or John. Hendricks work).
[1] W. S. Andrews, Magic Squares
and Cubes, Open Court Publishing, 1917, page 342.
[2] H. E. Dudeney, Modern Puzzles, 1926. This book has been included in
it’s entirety in 536 Puzzles & Curious Problems Charles Scribner’s Sons,
NY 1967 where this information appears in condensed form on pages 349350.
[3 ] Martin Gardner, Mathematical Recreations column of Scientific
American, Dec. 1965, reprinted with addendum in Martin Gardner,
Mathematical Carnival, Alfred A. Knoff, 1975
[4] J.R. Hendricks, The Aspiring Scientist, Unpublished, November, 1993,
page 10.
Complements
As mentioned previously, all basic solutions have a complement. The
complement of a star is found by subtracting each number from the sum of
the first and last number in the list of numbers used.
The complements are scattered throughout the ordered list of solutions
because the direct complement is an equivalent and must be normalized first,
before it can be found in the ordered list of basic solutions.
There are several coincidences worth noting.
 The first two solutions on the list are a complement pair.
Index numbers 5 and 6, 11 and 12, 31 and 32, 43 and 44 and 60, 61 are the only
other adjacent pairs.
The corresponding pair numbers are 1, 4, 9, 28, 34, and 40.
 The complements of solutions where the peaks are also magic have the
valleys magic. Two of these are adjacent, indices 38 and 39, with their
complements, the magic valleys, 79 and 78 also adjacent.
NOTE that normally peak and valley values are not complements of each other.
Adjacent
Complement Pairs
Solution numbers 
Pair # 
Remarks 
1 
2 
1 
The first pair number and the first two
solutions. 
5 
6 
4 
Pair # 2 consists of
solution 's 3 and 41 and pair # 3 consists of solutions 4 and 42.
Solutions 7 and 32 are pair # 5. In general, the second solution of
a pair are scattered throughout the list. 
11 
12 
9 
32 
33 
28 
The complement and pair
partner for each solution is shown in the
Order6 solution list. 
43 
44 
34 
60 
61 
40 
The last pair number consists of the
last adjacent pair of solutions. 
Twenty Sets of Four
The set of basic solutions may be divided into twenty sets of four
solutions each. These are shown in the table below, which also includes
the index number from the master list.
Each of twenty basic solutions may be changed into three other solutions by
the following transformations. These transformations will be equivalents and
must then be normalized to a basic solution. The solution sets are identified by
a number from 1 to 20, and the upper case letter A for the original with B, C
and D indicating the 3 transformations.

The first solution not yet the result of a
previous transformation.

Transformation of A by swapping b and d, e
and j, g and i, f and l.
1^{st} occurrence is index 12. A & B together are index numbers
1 to 40.

Transformation of A by swapping a and c, e
and g, h and k, i and j.
1^{st} occurrence is index 49.

Transformation of A by swapping a and h, b
and l, c and k, d and f.
1^{st} occurrence is index 41. C & D together are index numbers
41 to 80.


This feature was discovered by Mutsumi Suzuki.
[1]
Here of course, the solutions appear in my notation, i.e. line by line
rather then row by row.
Indx Main  Basic Solutions (Upper case = peaks) Index # of subset
# Set A b c D e f G h i J K L B C D
1 1A 1 2 11 12 3 5 6 10 9 8 4 7 19 67 43
2 2A 1 2 11 12 4 3 7 8 10 5 6 9 27 74 44
3 3A 1 2 12 11 3 4 8 7 10 5 6 9 31 66 41
4 4A 1 2 12 11 4 5 6 10 9 7 3 8 20 75 42
5 5A 1 3 10 12 2 4 8 6 11 5 9 7 30 49 61
6 6A 1 3 10 12 2 7 5 9 11 8 6 4 13 53 60
7 7A 1 3 12 10 4 7 5 11 9 8 2 6 16 73 59
8 8A 1 4 10 11 5 3 7 6 12 2 9 8 24 79 58
9 9A 1 4 11 10 2 9 5 8 12 7 6 3 12 52 72
10 10A 1 4 12 9 5 2 10 7 8 3 6 11 39 78 50
11 11A 1 5 8 12 3 2 9 6 10 4 11 7 37 62 48
14 12A 1 5 11 9 3 2 12 6 7 4 8 10 38 65 45
15 13A 1 5 11 9 3 8 6 12 7 10 2 4 23 64 77
17 14A 1 5 12 8 2 6 10 4 11 3 9 7 34 54 71
18 15A 1 5 12 8 7 2 9 10 6 4 3 11 35 80 47
21 16A 1 6 11 8 2 7 9 4 12 3 10 5 33 56 69
22 17A 1 6 12 7 3 5 11 4 10 2 9 8 29 68 57
25 18A 1 7 8 10 2 3 11 5 9 4 12 6 40 46 63
26 19A 1 7 8 10 4 3 9 5 11 2 12 6 36 76 55
28 20A 1 7 10 8 3 6 9 4 12 2 11 5 32 70 51
[1] Suzuki's site is now at
http://mathforum.com/te/exchange/hosted/suzuki/MagicStar.David.Prime.html.
Supermagic Stars
I refer to magic stars with the peaks also magic as SuperMagic Stars.
There are six of them as shown below , and their six complements have the
valleys magic.
Solutions 38 and 39 are adjacent peaks.
Two adjacent valleys are solutions 66 and 67, while numbers 77, 78, and 79 are
three adjacent valleys!
Indx comp. a b c d e f g h i j k l Magic
16 73 1 5 11 9 6 8 3 12 10 7 2 4 peaks
23 77 1 6 12 7 4 10 5 11 9 8 2 3 peaks
27 67 1 7 8 10 9 5 2 11 12 3 6 4 peaks
31 66 1 8 7 10 9 5 2 12 11 4 6 3 peaks
38 79 1 9 11 5 4 10 7 6 12 3 8 2 peaks
39 78 1 9 12 4 3 11 8 7 10 5 6 2 peaks
66 31 3 4 8 11 1 2 12 5 6 7 10 9 valleys
67 27 3 4 8 11 2 1 12 6 5 7 9 10 valleys
73 16 4 2 8 12 3 1 10 5 7 6 9 11 valleys
77 23 5 1 9 11 3 2 10 4 7 6 8 12 valleys
78 39 5 2 10 9 1 4 12 3 6 7 8 11 valleys
79 38 5 3 7 11 1 4 10 2 9 6 12 8 valleys
It is again worth noting that order6 is the only order that has solutions
with the peaks magic. For order7 the first 7 consecutive numbers equal 28
instead of 30 while all other consecutive sets are greater then 30.
With every ordern with n > 7, the number of first n
consecutive numbers is greater then the constant for that order.
All the magic peak solutions are type B of the 4 subsets of 20.
Of the six magic valleys, five are type C, the other is type D. Too bad!
Because the sum of the large triangle points are always equal, these sums will
each be 13 when peaks are magic and 26 when valleys are magic.
Cell
Pairs equaling 13
The magic hexagram may be divided into six pairs of two cells and then
classified into three groups on the basis of which pairs sum to 13. These pairs
are: A + C, B + D, E + I, F + L, G + J, and H + K.
This classification is similar to Dudeney’s, but involves all six cells.
Group H1 consists of 21 basic solutions with all 6 pairs
summing to 13.
Group H2 consists of 52 basic solutions with none of the six
pairs summing to 13.
Group H3 consists of 7 basic solutions. Pairs E + I and G + J both sum
to 13. The other 4 pairs do not sum to 13.
To put it another way:
if A + C = 13, then , B + D, F + L and H + K also = 13
if A + C ¹ 13, then , B + D, F + L and H + K
also ¹ 13
if E + I = 13 then G + J = 13: if E + I ¹
13 then G + J ¹ 13
if A + C = 13 and E + I = 13 then the hexagram is Group H1
if A + C ¹ 13 and E + I
¹ 13 then the hexagram is Group H2
if A + C ¹ 13 and E + I = 13 then the
hexagram is Group H3
The 28 Dudeney class 1c solutions consist of the
21 type H1 and the 7 type H3 groups.The group H2 solutions are
composed of all the
other Dudeney classes.
Six of the seven H3 solutions consist of three
complement pairs.


