6. THE CUBE
6.1 Introduction
A cube is a familiar object and gives us ability to think about three dimensional figures.
Consider the following example.
Illustration 1
A three centimeter cube has been painted red on all its sides. It is cut into one centimetre cubes.
How many cubes will be there with only one side painted red?
(A) 4 (B) 6 (C) 1 (D) 9 (E) None
Solution
It is clear that all cubes about the edges have either two or three sides painted.
Only one central cube in each face as shown in figure will have once face painted,
since all its other faces are hidden inside the large cube. The answer is 6.
Illustration 2
How many cubes have three sides pained?
(A) 4 (B) 6 (C) 8 (D) 12 (E) None of these
Solution
Each corner cube has three sides painted. There are in all eight corners,
hence the number is 8, i.e. the answer is (C).
Illustration 3
How many cubes have two sides painted?
(A) 4 (B) 6 (C) 8 (D) 12 (E) None of these
Solution
Each middle cube containing one of the edges has two sides painted.
There are twelve edges, hence there are 12 such cubes, i.e. the answer is (D)
Illustration 4
How many cubes have no side painted?
(A) 0 (B) 1 (C) 4 (D) 6 (E) 8
Solution
Only the innermost cube has no side painted, and in a 3 cm cube there
is only one such cube, i.e. the answer is (B).
EXERCISE-I
Directions: A solid cube painted red on two adjacent sides and black on the sides
opposite to the red sides and green on the remaining sides is cut into sixty four smaller cubes of equal size.
1. How many cubes are there which have no side painted?
(A) 0 (B) 4 (C) 8 (D) 16 (E) 24
2. How many cubes are there with three sides painted?
(A) 0 (B) 4 (C) 8 (D) 16 (E) 24
3. How many cubes have only one side painted?
(A) 48 (B)32 (C) 24 (D) 16 (E) 8
4. How many cubes have one or two sides but not three sides painted?
(A) 8 (B) 24 (C) 28 (D) 48 (E) 56
5. How many cubes have one side green and the adjacent side black or red?
(A) 8 (B) 16 (C) 24 (D) 28 (E) 36
Directions: A solid cube of 4 cm has been painted red, green and black on
pairs of opposite faces. It has then been cut into 1 cm blocks.
Questions 6-10 are with regard to smaller 1 cm cubes. Answer the following questions.
6. How many cubes have only one face painted?
(A) 0 (B) 8 (C) 16 (D) 24 (E) 32
7. How many cubes have only two faces painted?
(A) 0 (B) 8 (C) 16 (D) 24 (E) 32
8. How many cubes have only one three faces painted?
(A) 0 (B) 8 (C) 16 (D) 24 (E) 32
9. How many cubes have only four faces painted?
(A) 0 (B) 8 (C) 16 (D) 24 (E) 32
10. How many cubes have no face painted?
(A) 0 (B) 8 (C) 16 (D) 24 (E) 32
Directions: The six faces of a cube are painted in a manner that no two adjacent faces have the same colour.
The three colours used in painting are red, blue and green. The cube is then cut into 36 smaller cubes in such a
manner that 32 cubes are of one size and the rest of a bigger size and each of the bigger cubes has no red side.
11. How many cubes in all have a red side?
(A) 0 (B) 8 (C) 16 (D) 20 (E) 32
12. How many cubes in all have only one side coloured?
(A) 0 (B) 8 (C) 16 (D) 20 (E) 32
13. How many cubes are coloured on three sides?
(A) 0 (B) 8 (C) 16 (D) 20 (E) 32
14. How many cubes are there which have two or more sides painted?
(A) 36 (B) 32 (C) 28 (D) 20 (E) 16
15. How many cubes are there which are paianted on two sides only?
(A) 36 (B) 32 (C 28 (D) 20 (E) 16
Directions: In each of the questions 16-20 various faces of a cube are shown.
Study them and answer the questions by finding the correct alternative amongst the five given.
16. Which number is on the face opposite to 3?
(A) 1 (B) 2 (C) 4 (D) 5 (E) 6
17. Which number is on the face opposite to 6?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
18. Which number is on the face opposite to 6?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
19. Which number is opposite to 4?
(A) 1 (B) 2 (C) 3 (D) 5 (E) 6
20. Which number is opposite to 4?
(A) 1 (B) 2 (C) 3 (D) 5 (E) 6
A cube is paianted on all sides. Three adjacent sides are painted red.
Two of the remaining side are painted green and one is painted yellow.
The cube is then cut into eight small cubes. Answer the following questions
21. How many cubes are painted on more than three sides?
(A) 0 (B) 1 (C) 4 (D) 6 (E) 8
22. How many cubes are painted red on three sides?
(A) 0 (B) 1 (C) 4 (D) 6 (E) 8
23. How many cubes are painted red on two sides?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
24. How many cubes are painted red on one side?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 6
25. How many cubes have no side painted red?
(A) 0 (B) 1 (C) 2 (D) 4 (E) 6
Answers
Exercise-I
1. (C) 2. (C) 3. (C) 4. (D) 5. (B)
6. (D) 7. (D) 8. (B) 9. (A) 10. (B)
11. (E) 12. (B) 13. (B) 14. (C) 15. (D)
16. (C) 17. (D) 18. (A) 19. (A) 20. (A)
21. (A) 22. (B) 23. (D) 24. (C) 25. (B)
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