Evolution

The given system of equations can be rewritten as

$(x + 1)(y + 1) = 357, (y + 1)(z + 1) = 459$ and $(x + 1)(z + 1) = 567$ .

Thus $\dfrac{(x + 1)(y + 1)}{(y + 1)(z + 1)} \times (x + 1)(z + 1) = \dfrac{357}{459} \times 567 \Longrightarrow (x + 1)^{2} = 441 \Longrightarrow x + 1 = \pm 21$ .

As we are looking for positive $x,y,z$ we find that $x = 20$ . Next we have that

$y + 1 = \dfrac{357}{x + 1} = \dfrac{357}{21} = 17$ and that $z + 1 = \dfrac{567}{x + 1} = \dfrac{567}{21} = 27$ ,

and so $y = 16, z = 26$ . Finally, $(x + y - z)^{3} = (20 + 16 - 26)^{3} = 10^{3} = \boxed{1000}$ .