For those curious, I got Maple (a symbolic algebra solver thing) to solve the general case (resistors in the cube arrangement, but each resistor may have different, arbitrary values). I've attached a printout of the worksheet, but the TL;DR is:

The formula for the resistance is 3 pages long, and has this form (where capital letters are the various resistances, in an weird arrangement explained in the PDF):

total resistance = (A*U*G*F*C*V + K*B*A*E*C*F + ...) / (U*A*B*E*V + K*C*G*V*F + ...)

Which is at least dimensionally consistent!

In the case where four parallel resistors are equal (resistance = Rz) and the remaining 8 resistors are equal (resistance = Ry), then we find:

total resistance = (Ry^2 + 3 * Ry * Rz + Rz ^ 2) / (2 * Ry + 4 * Rz)

Notably, taking Ry or Rz to infinity brings the whole expression to infinity, as Ry and Rz both have higher powers in the numerator.

Finally, setting all the resistances to 1000 gives an answer of 833.33..., which is reassuring

Btw, hamster_nz, the above is not how I initially solved the simpler problem: I initially solved in exactly the way that Brumby described. Unfortunately, that method can't be applied in the case where the resistors have different values, as the underlying symmetry breaks down

I'd be fascinated to hear if anyone has a non-maching-requiring method of solving the arbitrary case; I'll be the first to admit that I have no idea how to do it by hand.