# CRE 4 - Coloring cube with multiple colors | LR - Cubes

**Answer the following questions based on the information given below.**

All the opposite faces of a big cube are colored with red, blue and green colors. After that, it is cut into 64 small equal cubes.

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there where one face is green and other one is either blue or red?

- (a)
28

- (b)
8

- (c)
16

- (d)
24

Answer: Option D

**Explanation** :

Here, we have 64 smaller cubes.

Since the smaller pieces are also cubes, this is possible only when equal number of pieces are obtained along each axis.

∴ 64 = 4 × 4 × 4

⇒ 4, 4, and 4 pieces along the three axes.

If we look at the right face, there are 12 cubes with green on one face and red/blue on the other.

Similarly, we will have 12 cubes with green and red/blue of the left face as well.

∴ Total cubes with green and blue/red = 12 + 12 = 24.

Hence, option (d).

Workspace:

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there whose no faces are colored?

- (a)
0

- (b)
4

- (c)
8

- (d)
16

Answer: Option C

**Explanation** :

Number of small cubes having no face colored = (n - 2)^{3} = (4 - 2)^{3} = 8.

Hence, option (c).

Workspace:

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there whose 3 faces are colored?

- (a)
4

- (b)
8

- (c)
16

- (d)
24

Answer: Option B

**Explanation** :

Three faces colored cubes are always at the corners.

Here since no two adjacent sides are of same color, the 8 corner cubes will have all three faces colored differently.

Hence, option (b).

Workspace:

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there whose only one face is colored?

- (a)
32

- (b)
8

- (c)
16

- (d)
24

Answer: Option D

**Explanation** :

Number of small cubes having only one face colored = 6(n - 2)^{2} = 6 × 4 = 24.

Hence, option (d).

Workspace:

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there whose at the most two faces are colored?

- (a)
48

- (b)
56

- (c)
28

- (d)
24

Answer: Option B

**Explanation** :

Number of small cubes having two faces colored = 12(n - 2) = 24, and

Number of small cubes having only one face colored = 6(n - 2)^{2} = 6 × 4 = 24, and

Number of small cubes having no face colored = (n - 2)^{3} = 8

∴ Total number of small cubes whose at the most two faces are colored = 24 + 24 + 8 = 56.

Hence, option (b).

Workspace:

**Answer the following questions based on the information given below.**

There is a cube in which one pair of adjacent faces is painted red; the second pair of adjacent faces is painted blue and a third pair of adjacent faces is painted green. This cube is now cut into 125 smaller but identical cubes.

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there with no red paint at all?

Answer: 80

**Explanation** :

Cubes with no red paint = Total cubes – cubes with red paint on them.

Total cubes = 125.

Number of cubes with red paint = 25 + 25 – 5 = 45 (25 cubes on both front and back face – 5 common cubes on the edge)

∴ Cubes with no red paint = 125 – 45 = 80.

Hence, 80.

Workspace:

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there with at least two different colors on their faces?

Answer: 35

**Explanation** :

All the edge (including corner) cubes will give us cubes with 2 faces painted.

Total edge cubes = 12n – 2 × 8 = 60 – 16 = 44.

Now, there will be 9 cubes, 3 each along the 3 edges which have same color both sides, which will have only 1 color on them.

∴ Number of cubes with two different colors on their faces = 44 – 9 = 35.

**Alternately,**

There are 9 edges with different colors on both sides each with 3 middle pieces.

∴ These 9 edges will give 9 × 3 = 27 cubes with 2 different colors on them.

There are also 8 corners all with 2 different colors on them.

∴ Number of cubes with two different colors on their faces = 27 + 8 = 35.

Hence, 35.

Workspace:

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there with one face painted red?

Answer: 40

**Explanation** :

If we look at the front surface in the diagram, there are 20 cubes with red on only one face.

(Edge cubes are excluded).

Similarly, there are 20 cubes on the right surface with red on only one face.

∴ Number of cubes with one face painted red = 20 + 20 = 40.

Hence, 40

Workspace:

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are with both red and green on their faces?

Answer: 13

**Explanation** :

There are 3 edges with red and green on either side.

∴ Number of cubes with red and green on their faces = 5 + 5 + 3 = 13.

Hence, 13.

Workspace:

**CRE 4 - Coloring cube with multiple colors | LR - Cubes**

How many small cubes are there showing only green or only blue on their faces?

Answer: 42

**Explanation** :

We get cubes with only one colour out of blue or green in 2 possible ways.

**Possibility 1:** The central 3 × 3 = 9 squares on a surface

There are 4 such surfaces (2 for blue and 2 for green)

∴ Cubes showing only green or blue = 4 × 9 = 36.

**Possibility 2:** The 3 middle-cubes along the common edge of two faces having the same colour.

There are 2 such edges (1 each for blue and green).

∴ Cubes showing only green or blue = 2 × 3 = 6.

∴ Total cubes showing only green or blue = 36 + 6 = 42.

Hence, 42.

Workspace:

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