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?
Answer & explanation
Answer: (a) 16
With pieces left unmoved, four plane cuts can give at most 15 pieces (1 + 4 + 6 + 4), so 16 is impossible. The other values arise easily from identical pieces after three cuts: 5 and 8 from four equal slabs, 12 from eight small cubes.
- Identical pieces after three cuts: 4 equal slabs (three parallel cuts), 6 equal blocks (two parallel + one perpendicular), or 8 small cubes (three mutually perpendicular cuts through the centre).
- Four plane cuts on an unmoved solid give at most 1 + 4 + 6 + 4 = 15 pieces; equivalently, the fourth plane can cross at most 7 of the 8 small cubes, giving 8 + 7 = 15.
- So x₄ = 16 is not possible.
- Check the rest: 4 slabs + a fourth parallel cut through one slab = 5; 4 slabs + a perpendicular cut through all four = 8; 8 cubes + a cut parallel to a face through four of them = 12.
- ✗ (a) Not possible: four plane cuts on unmoved pieces give at most 15 pieces.
- ✓ (b) Possible: after three central perpendicular cuts (8 cubes), a fourth cut parallel to one face through one layer splits 4 cubes → 12.
- ✓ (c) Possible: after three parallel cuts (4 equal slabs), a fourth cut perpendicular to them splits all 4 → 8.
- ✓ (d) Possible: after 4 equal slabs, a fourth cut parallel to them inside one slab → 5.
Remember · Unmoved pieces: plane cuts give at most 2, 4, 8 and 15 pieces for one to four cuts.
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·