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?

The correct option is A (16) because it is mathematically impossible to produce 16 pieces with 4 straight planar cuts on a solid cube without rearranging the pieces.

In discrete geometry, the maximum number of regions a 3-dimensional solid can be partitioned into by n planar cuts is described by the "Cake Number" sequence, a concept rooted in space-partitioning formulas developed by mathematician Jacob Steiner in 1826. The maximum number of pieces P is given by the formula P = (n³ + 5n + 6)/6. By substituting n=4, we find that the absolute maximum number of pieces is (64 + 20 + 6) / 6 = 15. Since the maximum possible number of pieces with 4 cuts is 15, obtaining 16 pieces is mathematically impossible for x₄.

The other options represent possible values for x₄, fulfilling the condition that the first 3 cuts produce identical pieces:

• Option B (12) is incorrect because 12 is a possible value: If the first 3 cuts are mutually perpendicular (one along the x, y, and z axes), they create 2 × 2 × 2 = 8 identical smaller cubes. A 4th cut parallel to one of the existing planes will cleanly slice exactly 4 of these cubes, adding 4 new pieces to yield 12 pieces total.
• Option C (8) is incorrect because 8 is a possible value: If the first 3 cuts are all parallel, they form 4 identical flat slices. A 4th cut perpendicular to them intersects all 4 slices, yielding 4 × 2 = 8 pieces.
• Option D (5) is incorrect because 5 is a possible value: If the first 3 cuts are parallel (yielding 4 identical slices), a 4th cut parallel to the first three adds an additional slice, yielding 4 + 1 = 5 pieces.

Takeaway: When solving 3D solid cutting puzzles where pieces remain unmoved, use the Cake Number sequence. The absolute maximum number of pieces for 1, 2, 3, and 4 planar cuts is 2, 4, 8, and 15, respectively.