QUESTION

CSAT

Hard

Maths

Prelims 2026

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 $x_4$ ?

Select an option to attempt

Explanation

Answer: D — 5

After three cuts producing 8 identical pieces, the cube has been split by mutually perpendicular cuts at $x=\tfrac{1}{2}$ , $y=\tfrac{1}{2}$ , $z=\tfrac{1}{2}$ , giving eight $\tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2}$ sub-cubes.

A 4th planar cut can only add pieces, never remove them. Each sub-cube the new plane passes through is split into exactly two pieces, so: $x_4 = 8 + (\text{number of sub-cubes the 4th plane intersects in its interior})$

This added quantity ranges from $0$ (plane coincides with a face/existing cut) up to $8$ (plane passes through the interior of every sub-cube).

Check the options.

$x_4 = 16$ : 4th plane through the interior of all 8 sub-cubes (e.g., a plane parallel to a face but offset by a quarter, plus one more parallel cut — geometric arrangements allow this). ✓ Possible.
$x_4 = 12$ : 4th plane cuts 4 sub-cubes (e.g., parallel to an existing cut, slicing one layer in half). ✓ Possible.
$x_4 = 8$ : 4th cut coincides with an existing cut, intersecting no sub-cube interior. ✓ Possible.
$x_4 = 5$ : would require fewer pieces than the 8 already present. A planar cut never reduces the number of pieces. ✗ Not possible.