Find the minimum surface area for a polycube composed of 2013 unit cubes.

You'd want to get as close to a large cube as possible. The 50 cubes could be arranged 3x4x4 with two additional adjacent cubes attached. The 3x4x4 has a surface of 4*4*2 + 4*3*2 + 4*3*2 = 80. The additional two cubes add only 6 unit squares as the two outer faces merely replace the two faces that get hidden. So the answer would be 86 for the 50-unit-cube case.

For the 2013-cube case, start with a 13x13x13 cube. You have an excess of 184, of which 169 could be taken care of by taking away one 13x13 plane making a 12x13x13 solid. If the remaining 15 were removed, you'd still have as much surface area exposed as the squares involved would merely be placed further in (assuming you take from the edges). The surface is then 12*13*2 + 12*13*2+13*13*2 = 962.

No ... wait ... A whole edge of 13 would be removed plus a couple more. So the two end squares of that row of 13 would disappear, leaving 960 instead of 962.