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: A — 16

The condition says that after the third cut, the pieces are identical. It does not necessarily mean that there are $8$ pieces. For example, three parallel equally spaced cuts can produce $4$ identical slabs, and other arrangements can produce $6$ or $8$ identical pieces.

A fourth cut can only increase the number of pieces by the number of existing pieces it actually cuts. Hence:

$x_4 = x_3 + \left(\text{number of existing pieces cut by the 4th plane}\right)$

Now check the options:

$x_4 = 5$ is possible: after three parallel cuts, there are $4$ identical slabs; the fourth cut may cut only one slab, giving $4+1=5$ .
$x_4 = 8$ is possible: after three parallel cuts, there are $4$ identical slabs; the fourth cut may cut all $4$ slabs, giving $4+4=8$ .
$x_4 = 12$ is possible: after three cuts producing $6$ identical pieces, the fourth cut may cut all $6$ pieces, giving $6+6=12$ .
$x_4 = 16$ would require $x_3=8$ and the fourth cut to cut all $8$ pieces. But after three cuts producing $8$ identical pieces, the cube is divided into a $2\times 2\times 2$ arrangement, and a single plane cannot pass through the interiors of all $8$ pieces.