Complete formula of 3x3 semi-magic squares of squares
by Lee Morgenstern, April 2008.
If this is any 3x3
semi-magic square of squares
A^2 B^2 C^2
D^2 E^2 F^2
G^2 H^2 I^2
r = (C - E) / (D - I)
s = (G - E) / (B - I)
u = (C - E) / (H - A)
v = (G - E) / (F - A)
as the slopes of lines going through pairs of points on hyperbolas.
a = 2r/(1-r^2), b = (1+r^2)/(1-r^2)
c = 2s/(1-s^2), d = (1+s^2)/(1-s^2)
e = 2u/(1-u^2), f = (1+u^2)/(1-u^2)
g = 2v/(1-v^2), h = (1+v^2)/(1-v^2)
we can describe the above semi-magic
I = E [e(h-d) - g(f-b)] / (ag-ce)
A = [E(f-b) + Ia] / e
D = Ea - Ib
C = Eb - Ia
B = Ec - Id
G = Ed - Ic
H = Ee - Af
F = Eg - Ah
Picking any rational values for r,s,u,v, and E and using the above formula always produces a semi-magic square of squares of rational numbers which can be scaled up to integers.
If the expressions for I and A are substituted into the rest, we can express all 9 entries from 4 terms r,s,u,v (and a...h).
E = ag - ce
I = bg - de - (fg - eh)
A = ah - cf - (ad - bc)
C = e(ad - bc) + a(fg - eh)
G = g(ad - bc) + c(fg - eh)
H = f(ad - bc) - a(fh - eg) + c
F = h(ad - bc) + c(fh - eg) - a
D = e(bd - ac) + b(fg - eh) - g
B = g(ac - bd) + d(fg - eh) + e
will produce a rational scaling of every possible 3x3 semi-magic square of squares.
r = 6/7
s = 1/2
u = 6/5
v = 3/2
produce the smallest 3x3 semi-magic square of squares
32^2 4^2 47^2
44^2 23^2 28^2
17^2 52^2 16^2
with magic sum = 3249.
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