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What is a Magic Hexagon?
The magic square first
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This is certainly well known:
You can distribute the numbers 1 to 9 in a 3x3 square in such way that
the sums vertically, horizontally and diagonally have the same value. It
is 15.
You find more on my page Magic Square.
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Correspondingly a magic hexagon is a figure, which contains the numbers
1 to 19 and where the sums horizontally (-), sloping up to the right (/)
and up to the left (\) are equal. It is 38.
There is only one hexagon, too.
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Searching for Magic Hexagons
top
Hexagons of the degree n
The magic hexagon is in a sequence of increasing hexagons.
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The hexagons have 1,7,19, 37, ... small hexagons.
Generally the hexagon of the order n has 3n²-3n+1 hexagons.
The hexagons have 1,3,5,7,..., generally
2n-1,
rows.
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Derivation of the
term 3n²-3n+1
If you calculate the differences of the numbers 1,7,19,37,..., you
get 6,12,18, ... and then as the difference of the differences 6, 6, 6,
... .
Thus you have a sequence of the form f(n)=an²+bn+c.
n=1 makes1=a+b+c
n=2 makes 7=4a+2b+c
n=3 makes 19=9a+3b+c |
The system of equations has the solution a=3, b= - 3 and c=1.
Thus the n-th term of the sequence is 3n²-3n+1. |
There is 3n²-3n+1=3n(n-1)+1. This
is the idea for the following geometric view.
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You can divide every hexagon in three parallelograms with the measurements
n and (n-1).
A small hexagon is left.
There is n=3 in the drawing. |
Searching for the
magic number
There are equal sums in a magic hexagon. This
sum H is called the magic number of the hexagon. If you add all numbers
in a hexagon, you have the sum S=H+H+...+H (z times) or S=zH. The variable
z is the number of parallel rows.
The number of rows is 1, 3, 5, 7, generally z = 2n-1.
According to the sum formula 1+2+3+...+m = m(m+1)/2 you get
S = (3n²-3n+1)[3(n+1)²-3(n+1)+1]/2
= (9n4-18n³+18n²-9n+2)/2.
Then you have H = S/z = S/(2n-1) = (9n4-18n³+18n²-9n+2)/[2(2n-1)]
= (9n4-18n³+18n²-9n+2)/(4n-2).
The magic number, given as a quotient, must be an integer.
You can show by a clever calculation that there is the only solution
n=3.
You recognize that H or 32H is only a whole number, if 5/(2n-1) is,
too. n=3 is a solution that fits. The numbers n= -2, 1,0 don't fit.
You can also find this solution by a computer. You replace n by "all"
numbers in the term H=(9n4-18n³+18n²-9n+2)/(4n-2)
and find the integers -2, 1,0, and 3.
Searching for the
places of the 19 numbers
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......
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You see that, if there is a magic hexagon, it must have the order 3.
It exists!
It is amazing that there is only one magic hexagon.
It isn't easy to find the distributions of the numbers 1 to 19. The
search is a combination of logic and trial.
You find descriptions at (3), (6), (Torsten Sillke, URL below).
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To the History
of the Magic Hexagon top
Martin Gardner made this topic
popular like he did with many problems lately.
He tells a nice story in book (4).
He heard of the magic hexagon by a letter sent by Clifford W.Addams,
a retired employee at a railway company from Philadelphia. This man started
in working on the hexagon in 1917 and found a solution in 1957 by trial
and error. He lost it and found it again in 1962.
Gardner highly regarded the problem, when he put the mathematician
Charles W. Trigg of the university of Los Angeles on it. He found out that
the magic hexagon is unique and obviously unknown in the mathematical literature.
Trigg published his proof in "Recreational Mathematics" (3) in 1964.
You can read his article at Torsten Sillke (URL below).
The problem was already known in the end
of the 19th century.
You can read at Harvey Heinz (URL below): "Jerry Slocum mailed me a
copy of an advertisement (?) dated 1896, crediting W. Radcliffe, Isle of
Man, U.K. with this discovery in 1895".
Heinrich Hemme published an article in
the magazine Bild der Wissenschaft (1988) and Hans F. Bauch in Wissenschaft
und Fortschritt (1990) that the royal architect (königlicher
Baumeister) Ernst von Haselberg from Stralsund knew, solved and proved
the definiteness of this problem in 1887 (1), (5), (6).
... ... |
This is a decoration wall beside the town hall of Stralsund, which
was renovated during "von Haselberg"'s period of office.
When I visited Stralsund in 1995 I thought it was worth taking a photo. |
Magic T-Hexagon top
There is another magic hexagon however formed
by triangles. It is called the magic T-hexagon (triangle hexagon). The
hexagon above is called H-hexagon (hexagon hexagon) in difference.
... ... |
The hexagon is an arrangement of 24
equilateral triangle.
You can distribute the numbers 1 to 24, so that it becomes magic.
12 sums horizontally (-), sloping up to the right (/) and up to the
left (\) are equal. It is 38.
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Number of the triangles
The magic hexagon is in a sequence of increasing hexagons.
... ...
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The hexagon of the order n has 6n² triangles.
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Magic numbers
It is S=zH again.
S = 1+2+3+ ... +6n² = 6n²(6n²+1)/2 = 18n4+3n²
and z = 2n makes
H = S/z = (18n4+3n²)/2n =[3n(6n²+1)]/2
The right hand term H is only then an integer, if the numerator is
even. The consequence is, n must be an even number.
Thus only the middle one of the three hexagons shown above can be magic.
There are many magic hexagons of this kind.
To the History of Magic
T-Hexagons top
... ... |
As far as known today the magic T-hexagon was investigated by Hans
F. Bauch for the first time. He published his results in the Mathematischen
Semesterberichten in
1991 (7).
The drawing is an example from the original version, a T-hexagon with
a simple inner hexagon.
The T-hexagon is called "magic hexagon D(4)". D is for Dreieck (triangle),
4 for the number of rows.
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You find the name T-Hexagon on John Baker's web site (URL below).
Obviously John Baker and David King found the T-Hexagons once more,
independent from Hans F. Bauch. You can read on John Baker's Site: "This
arrangement was discovered on 13th September, 2003 and as far as we can
ascertain is the first example of a magic T-hexagon."
Variants of the Magic
Hexagon top
1) Magic hexagon with integers
... ... |
You can find an arrangement of the successive numbers -4, -3, -2, ...,
13, 14 in a hexagon of the order 3, so that it is magic.
The magic number is 19. |
... ... |
This hexagon contains the numbers -9 to +9 with the magic
number 0.
You can probably generalize it: You can find an arrangement of the
numbers -3n(n-1)/2 to +3n(n-1)/2 in a hexagon of
the order 3, so that a magic hexagon with the magic number 0 develops.
(Information by Torsten Sillke, URL below).
There is no proof yet.
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You find magic hexagons of the orders 4, 5 and 7 at en.wikipedia.
They have start numbers >1. Zahray Arsen is the author.
2) Magic hexagon
and a mean
... ...
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The hexagon of the order 3 contains the numbers 1 to 19 like the magic
hexagon above.
All the 15 rows don't have the same sum, but the same mean 10.
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Origin: Fred W. Helenius ( Torsten Sillke)
3) Magic star
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It has the numbers 1 to 12.
There are 6 equal sums.
The magic number is 26.
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The magic star was known in the 19th century.
There are 96 arrangements in book (2).
You find more on the internet at "Suzanne
Alejandre and Mutsumi Suzuki's Magic Stars" (URL below)
4) Hexagram
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It has the numbers 1 to 12.
There are 6 sums with 4 summands.
The magic number is 33.
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Found at MathWorld (Bolt, B.; Eggleton, R.; and Gilks, J. "The Magic
Hexagram." Math. Gaz. 75, 141-142, 1991)
5) First figure of
9 hexagons
... ...
|
 ...
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If you take away 7 small hexagons at a hexagon of the order 4, you
get a figure with 3 concentric rings and a chain of 6 hexagons. |
... ... |
You can distribute the numbers 1 to 30 in this hexagon (2 views) in
such a way that it becomes magic.
There are 9 equal sums with 6 summands produced by the numbers in the
corners of the small hexagons.
The magic number is 93.
There are many solutions.
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Origin: Harvey Heinz (URL below)
6) Second figure
of 9 hexagons
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There is another arrangement of 9 hexagons, which you can fill with
the numbers 1 to 30, so that it becomes magic.
The figure is formed by 9 rings with common parts.
The magic number is 93. This is the sum of the numbers of one hexagon.
There are many solutions.
I found an information on Jaewook Shin's site with a picture as a reference:
"The original source of the problem: Gu-Su-Ryak by Choi, Seok Jung
(1646~1715). This book is displayed in the museum of Daejeon history in
Daejeon, Korea." |
Origin: Jaewook Shin (URL below)
The Magic Hexagon
on the Internet top
German
Wikipedia
Ernst von
Haselberg
English
David King
Hall of Hexagons
Eric W. Weisstein (World of Mathematics)
Hexagon,
Talisman
Hexagon,
MagicHexagon,
Hexagram
Frank R. Kschischang;
The
Magic Hexagon
Hans F Bauch
The
magic hexagon of Ernst v. Haselberg (.pdf-file)
Harvey D. Heinz
More Magic
Squares
Jaewook Shin
A
Numerical Solution of the Magic Hexagon Using Local Search and Global Search
Methods
Mutsumi Suzuki (bei mathforum)
Magic
Stars (recommended)
Torsten Sillke
Magic
Hexagon, Magic-Hexagon-Trigg
Wikipedia
Magic hexagon
References top
(1) Ernst von Haselberg:
Section 795: Zeitschrift für mathematischen
und naturwissenschaftlichen Unterricht 19 (1888) 429
Aufgabe
Section 801: Zeitschrift für mathematischen
und naturwissenschaftlichen Unterricht 20 (1889) 263-264 Auflösung
(2) Hermann Schubert: Mathematische Mußestunden, Neubearbeitung
von F.Fittig, Walter de Gruyter und Co, 1935
(3) Charles W. Trigg: A Unique Magic Hexagon, Recreational Mathematics
Magazine (January-February 1964)
(4) Martin Gardner: Mathematisches Labyrinth, Braunschweig/Wiesbaden
1979 [ISBN 3-528-08402-2]
(5) Heinrich Hemme: Das magische Sechseck, Bild der Wissenschaft (Oktober
1988) 164-166
(6) Hans F. Bauch: Zum magischen Sechseck von Ernst v. Haselberg, Wissenschaft
und Fortschritt 40:9 (1990)
(7) Hans F. Bauch: Magische Figuren in Parketten, Mathematische Semesterberichte
38:1 (1991)
Feedback: Email address on my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
2006 Jürgen Köller
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