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 xₙ. 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 x₄?
- (a) 16 ✓ UPSC's answer
- (b) 12
- (c) 8
- (d) 5
Why the answer is (a)
• With unmoved pieces, three cuts can give identical pieces in three ways: three parallel cuts (4 slabs), two parallel cuts and one across them (6 pieces), or three mutually perpendicular cuts through the centre (8 pieces).
• A fourth cut can then give 5 (another parallel cut on the 4 slabs), 8 (a cut across all 4 slabs) or 12 (a cut across all 6 pieces).
• 16 is impossible: four plane cuts can never make more than 15 pieces of a solid, and one plane cannot pass through all 8 pieces of the 2 × 2 × 2 arrangement.
• So 16 is not a possible value of x₄: (a).
Why the other options are wrong
- (b) 12
- 12 is possible: after 6 identical pieces, one more cut across all of them.
- (c) 8
- 8 is possible: after 4 parallel slabs, one cut across all four.
- (d) 5
- 5 is possible: after 4 parallel slabs, one more parallel cut.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.