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.
There is 3n²-3n+1=3n(n-1)+1. This is the idea for the following geometric view.
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
Then you have H = S/z = S/(2n-1) = (9n 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=(9n Searching for the places of the 19 numbers
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).
Number of the triangles The magic hexagon is in a sequence of increasing hexagons.
Magic numbers It is S=zH again. S = 1+2+3+ ... +6n² = 6n²(6n²+1)/2 = 18n ^{4}+3n²
and z = 2n makes
H = S/z = (18n ^{4}+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.
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."
There is no proof yet. 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
3) Magic star
You find more on the internet at "Suzanne Alejandre and Mutsumi Suzuki's Magic Stars" (URL below) 4) Hexagram
5) First figure of 9 hexagons
6) Second figure of 9 hexagons
German Wikipedia
English David King
Eric W. Weisstein (World of Mathematics)
Frank R. Kschischang;
Hans F Bauch
Mutsumi Suzuki (bei mathforum)
Torsten Sillke
Wikipedia
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