Integral Cuboid Problem

Question

Any progress in solving the integral cuboid problem?: Find a "brick" of integral dimensions a, b, c such that the 3 face-centered diagonals based on (a,b), (b,c), (a,c) AND the "body" diagonal based on (a,b,c) are all integral? Progress: theoretical or experimental.

Answer 1

Currently, using Solve, Reduce, or FindInstance, Mathematica can neither solve the problem nor prove that a solution does not exist.

Clear["Global`*"]

Format[d[s_List]] := Subscript[d, StringJoin[ToString /@ s]]

sides = {a, b, c};

eqns = Assuming[Thread[sides > 0],
   d[#] == (Norm[#] // Simplify) & /@
    Subsets[sides, {2, 3}]];

Solve[eqns, sides, Integers]

Reduce[eqns, sides, Integers]

FindInstance[eqns, sides, Integers]

EDIT: Equivalent results are obtained after adding inequalities to the equations.

eqns2 = Module[{ss = Subsets[sides, {2, 3}]},
  Join[
   Assuming[Thread[sides > 0],
      d[#] == (Norm[#] // Simplify)] & /@ ss,
   Thread[sides > 0], Thread[(d /@ ss) > 0]]]

Solve[eqns2, sides, Integers, MaxExtraConditions -> All]

Reduce[eqns2, sides, Integers]

FindInstance[eqns2, sides, Integers]