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 $x_n$. 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 $x_4$?
Detailed Explanation:
After three cuts producing identical pieces, the fourth cut increases the total by the number of pieces it intersects: x₄ = x₃ + (pieces cut by 4th plane).
For x₄ = 5: Three parallel cuts yield 4 identical slabs; the 4th cut intersects 1 slab → 4+1=5. For x₄ = 8: Same 4 slabs, 4th cut intersects all 4 → 4+4=8. For x₄ = 12: Three cuts yield 6 identical pieces; 4th cut intersects all 6 → 6+6=12. For x₄ = 16: Requires x₃=8 (a 2×2×2 cube arrangement) and the 4th cut intersecting all 8 pieces. However, a single plane cannot pass through the interiors of all 8 corner cubes in a 2×2×2 arrangement simultaneously.
Therefore, 16 is geometrically impossible, making Option 1 the correct answer.
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