There Can't Be Too Many?

An \(x \times y \times z\) rectangular prism is made out of \(xyz\) identical unit cubes. Divide these cubes into 2 mutually exclusive sets.

The first set contains all cubes which have at least one face that is a constituent of the exterior faces of the rectangular prism. These cubes will be called external.

The second set contains all cubes which are completely enclosed inside the rectangular prism. These cubes will be called internal.

An example is that in a \(3 \times 4 \times 7\) rectangular prism, there are 74 external cubes and 10 internal cubes.

How many different rectangular prisms are there, whose number of internal cubes is equal to the number of external cubes?

If you think the answer is infinite, type -1.

Details and Assumptions:

  • Rotations/Flips among any axis are considered the same.

  • Rectangular prisms can have square faces

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