A cube of integer side length a is made of unit cubes. Write an expression giving the number of unit cubes to be added to make a cube of side length a + 1.
Answer: (a + 1)3 − a3 = 3a2 + 3a + 1 unit cubes.
Step-by-step solution
Idea: The cube of side a has a3 unit cubes; the cube of side a + 1 has (a + 1)3. The difference is what must be added.
- Cubes needed = (a + 1)3 − a3 = (a3 + 3a2 + 3a + 1) − a3.1 mark
- = 3a2 + 3a + 1. (Picture it: three a × a slabs on three faces, three rods of a cubes along the edges between them, and one corner cube.)1 mark
3a² + 3a + 1 unit cubes must be added.
Check: a = 2: 33 − 23 = 27 − 8 = 19 and 3(4) + 3(2) + 1 = 19 ✓.
Answer to write in the exam
Cubes to add = (a + 1)3 − a3
= a3 + 3a2 + 3a + 1 − a3
∴ 3a2 + 3a + 1 unit cubes
Common mistakes that cost marks
- Answering 1 or a + 1 (thinking only one layer or one row is added).
- Expanding (a + 1)3 as a3 + 1.
How this can come in the exam
MCQ (1 mark)
How many unit cubes must be added to a 4 × 4 × 4 cube to make a 5 × 5 × 5 cube?
- 25
- 61
- 64
- 125
Show answer
(B) 61
125 − 64 = 61 = 3(16) + 3(4) + 1.
Try one yourself
How many unit cubes must be removed from a cube of side a + 1 to leave a cube of side a − 1? Check for a = 3.
Show answer
(a + 1)3 − (a − 1)3 = 6a2 + 2; for a = 3: 64 − 8 = 56 = 54 + 2 ✓.
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