A cut on a solid object divides the object into two parts where the new surfaces thus produced are plane. On the other hand, one single cut can be used to cut more than one object at a time. In an experiment, the total number of pieces produced by applying n cuts is denoted by xn. The experiment is performed on a solid cube where pieces remain unmoved after each cut. In this experiment, if after the third cut, the pieces are identical, then which of the following is not a possible value for x4?

• A single planar cut on a solid object can increase the total number of pieces by at most the number of existing pieces it intersects, but practically, for a cube, the maximum pieces after $n$ cuts is given by the lazy caterer's sequence or similar spatial partitioning limits, specifically $x_n \\le x_{n-1} + n$ is a common lower bound for distinct planes, but here we look at specific configurations.

• After 3 cuts, the problem states the pieces are identical. The only way to divide a cube into identical pieces with 3 planar cuts is by making 3 mutually perpendicular cuts through the center, resulting in 8 identical smaller cubes ($x_3 = 8$).

• The 4th cut is a single plane. To maximize the number of new pieces, this plane must intersect as many of the existing 8 small cubes as possible.

• A single plane can intersect at most 4 of the 8 small cubes formed by the central cross (since the 8 cubes are arranged in a 2x2x2 grid, a plane can pass through at most 4 of them, e.g., a diagonal plane or a plane parallel to one of the original cuts but shifted, though shifting doesn't help intersect more if it's a single continuous plane across the whole assembly). Actually, a plane can intersect at most 4 of the 8 octants if it passes through the center, or potentially more if it slices through the corners? No, a plane divides space into two half-spaces. The 8 small cubes are in 8 octants. A plane can intersect at most 4 of these 8 cubes if it passes through the origin (center). If it does not pass through the origin, it can intersect at most 4 cubes as well (e.g., slicing off a corner layer). Wait, let's re-evaluate. The maximum number of pieces a single cut can add is equal to the number of existing pieces it cuts through. If the 4th cut passes through the center, it cuts through 4 of the 8 small cubes (the ones it bisects). This adds 4 new pieces. Total $x_4 = 8 + 4 = 12$.

• If the 4th cut is parallel to one of the previous cuts and passes through the center, it coincides with an existing cut or is redundant? No, if it's a new cut, it must be distinct. If it is parallel to one of the first three and passes through the center, it is the same plane. So it must be distinct. If it is distinct and parallel, it cannot pass through the center. It can cut through at most 4 of the 8 small cubes (e.g., slicing the top layer). This adds 4 pieces. Total 12.

• Can it cut through more? A plane can intersect at most 4 of the 8 sub-cubes in a 2x2x2 arrangement. Therefore, the maximum increase is 4. So $x_4$ can be at most $8+4=12$.

• Thus, 16 is not a possible value for $x_4$.