I stumbled upon this puzzle when working on a problem for fun on my spare time and I seemed to hit a wall. (Not gonna reveal what problem it is just yet, should be a pleasent surprise if it gets solved :) )
What I noticed is the following:
If you have 2 three square sum solutions to a number $r^{2}$ (r $\in \mathbb{R}$):
${x_1}^{2}+{y_1}^{2}+{z_1}^{2} = {x_2}^{2}+{y_2}^{2}+{z_2}^{2} = r^{2}$
the two following conditions cannnot be true at the same time:
$x_1 > x_2 \ge y_2 > y_1 \ge z_2 > z_1 $
$x_1 -x_2 > y_2 - y_1 > z_2 - z_1$
$x_1,x_2,y_2,y_1,z_2,z_1$ can all be real numbers as well.
Using non tradition spherical coordinates (lattitude and longitude):
$ x_1 = r\cdot cos \phi_1 \cdot cos \lambda_1 \\ y_1 = r\cdot cos \phi_1 \cdot sin \lambda_1 \\ z_1 = r\cdot sin \phi_1 $
$ x_2 = r\cdot cos \phi_2 \cdot cos \lambda_2 \\ y_2 = r\cdot cos \phi_2 \cdot sin \lambda_2 \\ z_2 = r\cdot sin \phi_2 $
I was able to show the following statements about the angles:
$\lambda_2> \lambda_1 > \phi_2 > \phi_1 $
$\lambda_2 - \lambda_1 > \phi_2 - \phi_1$
$sin \lambda_2 - sin \lambda_1 > sin \phi_2 - sin \phi_1$
And that all angles must be smaller than $\frac{\pi}{4}$.
As well as:
$\lambda_1 > \frac{\pi}{8} + \frac{\lambda_1 - \phi_2}{2}$
But still seeking the contradiction some new ideas are here needed! Or maybe this was already done and someone can give me a link to a paper? (I did search for it for a long time and found nothing). Thanks in advance!!
For this type of equations - which has something in common...
$$A^2+B^2+C^2=X^2+Y^2+Z^2=Q^2$$
Record regarding $Q$ - A different equation. And replace solving a system of equations, the solution of the same equation.
We write as....
$$A=2ac$$
$$B=2bc$$
$$C=a^2+b^2-c^2$$
$$Q=a^2+b^2+c^2=x^2+y^2+z^2$$
$$X=2xz$$
$$Y=2yz$$
$$Z=x^2+y^2-z^2$$
And instead of solving the system, we solve it.
$$Q=a^2+b^2+c^2=x^2+y^2+z^2$$
You can write a simple solution:
$$a=q^2+s^2+k^2-p^2-t^2$$
$$b=q^2+s^2+k^2-p^2+t^2+2pt-2kt-2st-2qt$$
$$c=q^2+s^2+k^2+p^2-t^2+2tp-2kp-2sp-2qp$$
$$x=p^2+t^2+k^2-q^2-s^2-2kt-2pk+2sk+2qk$$
$$y=p^2+t^2-k^2-q^2+s^2-2ts-2ps+2ks+2qs$$
$$z=p^2+t^2-k^2+q^2-s^2-2tq-2pq+2kq+2sq$$
$p,t,k,q,s$ - any integer asked us.
A similar method can be used for other type of equations.