UPSC Prelims 2026 CSAT, question 24

Topic: Geometry and mensuration

Q24

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 $x_4$ ?

(a) 16Correct answer
(b) 12
(c) 8
(d) 5

Solution.

Answer: (a) 16

Four plane cuts of a solid, pieces left in place, give at most $\binom{4}{0} + \binom{4}{1} + \binom{4}{2} + \binom{4}{3} = 1 + 4 + 6 + 4 = 15$ pieces, so $x_4 = 16$ is impossible.
The other values occur: three parallel cuts give 4 identical slabs, and a fourth cut through one slab or across all four gives $x_4 = 5$ or $8$ .
Three mid-plane cuts give 8 identical small cubes, and a fourth cut through 4 of them gives $x_4 = 12$ .