So you’d like to build some Magical Hexagons, but don’t know where to start?
This page will introduce to you some basic terminology, which, in combination with
several formulas, will allow you to determine the basic properties of the Hexagon that you wish
to develop.
First of all, here is some basic notations which I use to designate the characteristics of a
Magical Hexagon:
1) c_R, n, N : Any Regular hexagon consists of rings. Starting from any corner of your Hexagonal structure, and counting
the number of cells from that corner to the central cell, inclusively, will determine the Number of rings that
Hexagon contains. This Number I designate as c_R. Also I might refer to it as n or N, as in
“N Ring Magical Hexagon”
2) c_C : The Number of individual cells which a Hexagonal Structure totally contains is c_C
3) c_Rows : This is the number of rows along any one “side” of the overall Hexagonal structure. It also is identical to
the length of any of the Major Axis.
4) mNum : The sum of any row of the hexagonal structure adds to this number.
5) a_N : The Average number if you sum all the cell numbers together and divide by c_C.
Also, The range of integral numbers used in the hexagonal structure are designated as LowNum and HighNum, where LowNum < HighNum.
Now, the following are basic equations that determine some of the properties of the hexagonal structure.
c_C = 3*c_R2 – 3*c_R + 1
HighNum = LowNum + c_C – 1
c_Rows = 2*c_R – 1
a_N = (LowNum + HighNum)/2
mNum = a_N*c_C/c_Rows
Also:
c_C = (3*c_Rows2 + 1)/4
LowNum = a_N – (c_C – 1)/2
HighNum = a_N + (c_C – 1)/2
c_R = (c_Rows + 1)/2
Some more advanced equations are:
For sum integral value r, where r >= 0, then:
a_N = r*c_Rows
mNum = r*c_C
LowNum = r*c_Rows – (c_C – 1)/2
HighNum = r*c_Rows + (c_C – 1)/2
c_Rows = [4r +/–u]/3 , Where u2 = 16r2 – 3*(8*LowNum – 3, or rearranged, (4r-u)*(4r+u) = 3*(8*LowNum – 3)
Now, how would you use these equations?
Example #1) What are some possible mNums and their corresponding (LowNum, HighNum) Pairs for 3 ring hexes?
- From this question, we see that c_R = 3. thus some basic properties
o From c_C = 3*c_R2 – 3*c_R + 1, we see that c_C = 19.
o c_Rows = 2*3 –1, or 5
- an N ring where N >= 3 can have multiple ranges and Magical Sums. Thus, the index r is used to
determine these properties.
o when r = 0, mNum is always 0, and (LowNum, HighNum) always = (-(c_C-1)/2, (c_C-1)/2)
o As r incriments by +1, mNum incriments by c_C, and LowNum, HighNum incriments by c_Rows.
o So, specifically for a 3Ring, for r>=0:
§ mNum = 19r or mNum = 0, 19, 38, …
§ LowNum = 5r – 9 or LowNum = -9, -4, 1, …
§ HighNum = 5r + 9 or HighNum = 9, 14, 19,…
Example #2) What are the possible c_R and mNum pairs of all the mHexes whose LowNum = 2?
- First we have to determine those sets of (4r-u), (4r+u) where at least one of them are divisble by 3.
- Starting with (4r-u)*(4r+u) = 3*(8*LowNum – 3):
- (4r-u)(4r+u) = 3*(8*2 – 3) è (4r-u)(4r+u) = 3*13 è (4r-u)(4r+u) = 39
o 39 = 1*39, so lets say 4r-u = 1, and 4r+u = 39
§ Since 4r+u is divisible by 3, we can continue and simultaneously solve for r and u:
· (4r-u) + (4r+u) = 40 è 8r = 40 or
o r = 5, {so u must be 19}
· with c_Rows = [4r +/–u]/3, then c_Rows = (4r+u)/3 è c_Rows = 39/3, or:
o c_Rows = 13
· Then c_R = (13+1)/2, or:
o c_R = 7
· c_C = 3*(72)-3*7+1, or:
o c_C = 127
· from mNum = r*c_Cè mNum = 127*5, or:
o mNum = 635
· From HighNum = r*c_Rows + (c_C – 1)/2è HighNum = 5*13 + ((127-1)/2) or
o HighNum = 128
o 39 = 3*13, so lets say 4r-u = 3, and 4r+u = 13
§ Since 4r – u is divisible by 3, we can continue and simultaneously solve for r and u:
· r = 2, u = 5
· with c_Rows = [4r +/–u]/3, then c_Rows = (4r-u)/3 è c_Rows = 3/3, or:
o c_Rows = 1
· Then c_R = (1+1)/2, or:
o c_R = 1
- So, the only possible Magical Hexagons that can start with LowNum = 2 are 7 Rings Magical Hexagons,
where r = 5 and mNum = 635, and the trivial 1 ring: