For information, here is a small OPL model using CP Optimizer to solve the IBM Research Ponder This challenge of July 2013.

Problem:

When you roll three octahedron (eight sided) dice with the sides numbered 1, 4, 16, 64, 256, 1024, 4096, 16384, there are 120 different sums which can be produced. The maximal value, in this example, is 16384.

Find the eight positive integers that could number a octahedral die and minimize the maximum value of the eight faces, while still resulting in 120 possible sums when three of the dice are rolled.

As you see, the model is very short. CP Optimizer (V12.5.1) finds a solution for maximum value 300 in about 4mn on my laptop. Proving the optimality of this solution takes longer but CP Optimizer can make it.

Here is the complete model:

using CP;

int vmin = 1;

int vmax = 130; // 129 => Infeasible for vmax=129

int nfaces = 8;tuple Combi { int v1; int v2; int v3; }

{Combi} Combis = { <v1,v2,v3> | v1,v2,v3 in 1..nfaces : v1<=v2 && v2<=v3 };assert(card(Combis) == 120);

dvar int v[1..nfaces] in vmin..vmax;

dvar int total[c in Combis] in 3*vmin..3*vmax;execute {

cp.setSearchPhases(cp.factory.searchPhase(v));

cp.param.AllDiffInferenceLevel = "Extended";

};minimize max(i in 1..nfaces) v[i];

subject to {

v[1]==1;

forall(i in 2..nfaces) { v[i-1] < v[i]; }

forall(c in Combis) { total[c] == v[c.v1]+v[c.v2]+v[c.v3]; }

allDifferent(total);

}

And here is an optimal solution: v = [ 1, 3, 6, 35, 75, 108, 121, 130 ]

Note that this model can be made more efficient by adding surrogate constraints but we wanted to share the simplicity of the basic model.

Philippe Laborie and Paul Shaw