<s>[INST] Does this equation have positive integer solutions? The only solution I can find for $a^2 + b^2 + c^2 = d^2 + e^2 + f^2$   is $a=0$, $b=0$, $c=0$, $d=0$, $e=0$, $f=0$.  Are there any positive integer solutions?  Any where none of $a,b,c,d,e,f$ are equal?
 [/INST] Solving $a^2 + b^2 + c^2 = d^2 + e^2 + f^2$ has many many (boring) positive integer solutions.
Choose any bijection  $\theta: \{a,b,c\} \to \{d,e,f\}$. Then, choose any positive integer values for $\{a,b,c\}$. 
It must be true that $a^2 + b^2 +c^2 = \theta(a)^2 + \theta(b)^2 + \theta(c)^2 = d^2 + e^2 + f^2.$ As stated though, these are all pretty boring, and probably not quite what you're looking for. For less "boring" solutions, combining some known Pythagorean triples could help. 
For example, $3^2 + 4^2 = 5^2$. Meanwhile $8^2 + 15^2 = 17^2$ (I think, somebody else can fact-check this triple). Thus,
$(3^2+4^2)+17^2 = 5^2 + (8^2+15^2)$
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