Set of numbers from $1-9$ multiplied together to get the smallest possible value

Question

The numbers $x_1,$ $x_2,$ $x_3,$ $y_1,$ $y_2,$ $y_3,$ $z_1,$ $z_2,$ $z_3$ are equal to the numbers $1,$ $2,$ $3,$ $\dots,$ $9$ in some order. Find the smallest possible value of
$$x_1 x_2 x_3 + y_1 y_2 y_3 + z_1 z_2 z_3.$$

I would assume the lowest number, $1,$ would have to be multiplied by $9,$ the highest. I do not know how to approach this with AM-GM, though.

Answer 1

The following Mathematica script confirms that there is only one solution:

prod[p_] := p[[1]] p[[2]] p[[3]] + p[[4]] p[[5]] p[[6]] + p[[7]] p[[8]] p[[9]];
perms = Permutations[{1, 2, 3, 4, 5, 6, 7, 8, 9}];
unique = Select[perms, (#[[1]] < #[[2]] < #[[3]] && #[[4]] < #[[5]] < #[[6]] && #[[7]] < #[[8]] < #[[9]] && #[[1]] < #[[4]] < #[[7]]) &];
products = Map[prod, unique];
min = Min[products];
result = Select[unique, (prod[#] == min) &];
Print[min, " ", result]

The script prints:

214 {{1, 8, 9, 2, 5, 7, 3, 4, 6}}

The next best two solutions are:

215 {{1, 7, 9, 2, 5, 8, 3, 4, 6}}
216 {{1, 8, 9, 2, 5, 6, 3, 4, 7}, {1, 8, 9, 2, 6, 7, 3, 4, 5}}