Magic Hexagon of Triangles
Derivation of patterns:-
Here are a number of results which hold for all these magic hexagons:-
p+q+r+a+b=75; d+e+f+g+h=75; j+k+l+m+n=75;
whole hexagon is 4 rows x 75;
therefore c+s+t+u+o+x+w+v+i=75:
c+s+t+u+o+x+w+v+i=75;
r+s+t+u+f+x+w+v+l=75;
their intersection is s+t+u+x+w+v;
therefore r+f+l=c+o+i:
p+q+r+a+b=75=a+b+c+d+e;
therefore p+q+r=c+d+e:
(p+q+r+a+b) + (n+o+x+s+t+c) = 150;
(b+c+t+u+v+i+j) + (d+e+f+g+h) = 150;
(q+p+o+n+m) + (a+r+s+x+w+l+k) = 150;
(m+l+w+v+u+f+e) + (k+j+i+h+g) = 150;
therefore b+c+d+t = m+l+k+w:
(a+b+c+d+e) + (q+r+s+t+u+f+g) = 150;
b+c+d+t = m+l+k+w;
therefore (a+q+r+s) + (e+u+f+g) + (w+m+l+k) = 150:
(a+q+r+s) + (e+u+f+g) + (w+m+l+k) = 150 (above);
r+f+l=c+o+i (above);
therefore a+c+e+q+s+u+g+o+w+i+m+k = 150:
a+c+e+q+s+u+g+o+w+i+m+k = 150;
(a+b+c+d+e) + (q+r+s+t+u+f+g) = 150;
therefore b+d+r+t+f=o+w+i+m+k:
(a+b+c+d+e) + (q+r+s+t+u+f+g) = 150;
(p+o+x+w+v+i+h) + (n+m+l+k+j) = 150;
b+d+r+t+f=o+w+i+m+k (above);
therefore a+c+e+q+s+u+g=p+x+v+h+n+l+j:
b+c+t+u+v+i+j=75;
n+o+x+s+t+c+d=75;
c+t are common,
therefore b+u+v+i+j=n+o+x+s+d
b+a+r=o+n+m from above;
also n+m+l=g+h+i;
so b+a+r + n+m+l=o+n+m + g+h+i
therefore b+a+r+l=g+h+i+o:
r+s+t+u+f+wx+w+v+l=75 from above;
q+r+s+t+u+f+g=75;
so r+s+t+u+f+wx+w+v+l=q+r+s+t+u+f+g;
therefore x+w+v+l=q+g:
x+w+v+l=q+g from above;
so c+s+t+u=p+h;
therefore x+w+v+l + p+h = c+s+t+u + q+g:
x+w+v+l + p+h = c+s+t+u + q+g above;
b+c+d+t = m+l+k+w also above;
adding these, x+w+v+l + p+h + b+c+d+t = c+s+t+u + q+g + m+l+k+w;
therefore b+d + p+x+v+h = m+k + q+s+u+g:
b+d + p+x+v+h = m+k + q+s+u+g above;
which is (75-a-c-e) + (75-o-w-i) = (75-n-l-j) + (75-r-t-f)
therefore n+l+j + r+t+f = a+c+e + o+w+i:
n+l+j + r+t+f = a+c+e + o+w+i above;
r+f+l=c+o+i also above;
therefore n+j+t=a+e+w:
n+j+t=a+e+w above;
r+f+l=c+o+i also above;
therefore n+j+t + c+o+i = a+e+w + r+f+l:
n+j+t=a+e+w from above;
rotating, b+p+v=s+g+k;
again rotating, q+m+u=d+h+x;
therefore b+p+v + q+m+u = s+g+k + d+h+x:
a+r+s+x+w+l+k + q+p+o+n+m = 150
p+q+r+a+b = 75
therefore s+x+w+l+k+o+n+m = 75+b
Blue region adds up to (75 + green region)
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