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 $x_4$?

The correct option is (a) - 16.

[as per provisional answerkey]

Solution

The problem states that after 3 cuts, the pieces are identical and unmoved. For a cube to be divided into identical pieces without moving them, the cuts must be made along the three axes (X, Y, and Z) or parallel to each other.

Step 1: Analyze the state after 3 cuts (x₃)
To get identical pieces after 3 cuts without moving them, there are two primary configurations:
1. Parallel cuts: All 3 cuts are in the same direction. This results in $3+1=4$ identical rectangular slabs. (x₃ = 4)
2. Mutually perpendicular cuts: 1 cut along the X-axis, 1 along the Y-axis, and 1 along the Z-axis. This results in $2\times 2\times 2=8$ identical smaller cubes. (x₃ = 8)

Step 2: Calculate possible values for x₄ (after the 4th cut)
The 4th cut is applied to the existing configuration.
Case A: If x₃ = 4 (4 slabs)
- If the 4th cut is parallel to the first three, it divides all 4 slabs, resulting in $4+1=5$ pieces.
- If the 4th cut is perpendicular to the first three, it intersects all 4 slabs, resulting in $4\times 2=8$ pieces.
Case B: If x₃ = 8 (8 small cubes)
- If the 4th cut is parallel to any of the previous cuts (e.g., a second cut on the X-axis), it will pass through all existing pieces. Since we have a 2x2x2 grid, a new plane parallel to one axis will intersect 4 pieces. However, the question states "one single cut can be used to cut more than one object at a time." A single plane cut through the 2x2x2 arrangement will bisect 4 pieces, adding 4 new pieces. Total = $8+4=12$ pieces.

Conclusion: The possible values for x₄ are 5, 8, and 12. Therefore, 16 is not possible.

Why the other options are incorrect

• Option (b) - 12: This is a possible value. If the first 3 cuts are mutually perpendicular (creating 8 pieces in a 2x2x2 grid), a 4th cut parallel to any of the existing planes will slice through 4 of the small cubes, adding 4 more pieces ($8+4=12$).
• Option (c) - 8: This is a possible value. If the first 3 cuts are parallel (creating 4 slabs), a 4th cut perpendicular to them will bisect all 4 slabs, resulting in $4\times 2=8$ pieces.
• Option (d) - 5: This is a possible value. If all 4 cuts are made parallel to each other in the same direction, the number of pieces is simply n + 1, which is $4+1=5$.

Key Concept

The maximum number of pieces produced by n cuts in a cube (where pieces are not rearranged) is determined by the distribution of cuts across the three axes ($p+q+r=n$), but the constraint of "identical pieces after 3 cuts" limits the starting configuration for the 4th cut.