DSSSB TGT CS Mock 03 – Question 007

A cube is painted red on two opposite faces, green on two adjacent faces, and blue on the remaining two faces. It is then cut into 125 smaller cubes of equal size. How many smaller cubes have at least one red and at least one green face?

• 16
• 12
• 18
• 14

For a cube of side \(n = \sqrt[3]{125} = 5\):
– Let the 6 faces be: Top & Bottom = Red (Opposite).
– Front & Right = Green (Adjacent).
– Back & Left = Blue (Remaining).
Cubes with both Red and Green will lie along the edges joining a Red face and a Green face, plus the corner cubes joining them.
Red faces: Top, Bottom.
Green faces: Front, Right.
Edges shared between Red and Green:
1. Top-Front edge
2. Top-Right edge
3. Bottom-Front edge
4. Bottom-Right edge
Total such edges = 4 edges.
On each edge of length \(n = 5\):
– Middle edge cubes = \(n – 2 = 3\) cubes.
– Corner cubes sharing Top-Front: Top-Front-Right (Red, Green, Green) and Top-Front-Left (Red, Green, Blue).
Let’s count all cubes on these 4 edges:
– Each edge has 5 cubes. 4 edges share the Top-Front-Right corner and Bottom-Front-Right corner.
Using Inclusion-Exclusion for the 4 edges:
\(4 \times 5 – 2 \times 2 = 20 – 4 = 16\) cubes.
All these 16 cubes have at least one Red face and at least one Green face.