The solution (or actually one of many solutions) of this cubic equation with minimal integer positive value of (x+y+z), is x = 30,864,197 , y = -246,913,580 , z = 216,049,384 ; in that case, x+y+z = 1, and you can't get smaller than that. (Sorry, I wrote the same thing as a remark in the solution script, but I made a typo and wrote the wrong value of z there.)
Right and wrong
Back to basics 9 - Indexed References
Getting logical indexes
Sum the entries of each column of a matrix which satisfy a logical condition.
Rubik's Mini-Cube: Solve Randomized Mini-Cube - Score : Moves
Creation of 2D Sinc Surface
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