Question

a cube (made up of 27 smaller cubes) is painted on the outside. How Many of the smaller cubes would be painted on:
a) 3 faces
b) 2 faces
c) 1 face
d) no faces

Answer 1

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A painted cube that is cut into 27 equal-size smaller cubes has been cut into a 3 * 3 * 3 arrangement.

There is 1 cube in the very center (middle of 3 in each axis direction -- pitch, roll, and yaw), so 1 cube has no paint.

On each of the 6 sides of the cube, there is a central smaller cube that is painted once.

Also on each of the 6 sides of the cube, there are 4 cubes (at the middle of the edges). These are shared with one other side of the cube (or we could just count the 12 edge lines -- 12). So, 6 * 4 / 2, or just 12, smaller cubes are painted on 2 edges.

That leaves the 8 corners of the original cube, which are painted on 3 surfaces.

1 (no paint) + 6 (painted 1) + 12 (painted 2) + 8 (painted 3) = 27 total

Answer 2

a) 8 cubes would have 3 faces painted on.

b) 12 cubes would have 2 faces painted on.

c) 6 cubes would have 1 face painted on.

d) 1 cube would have no faces painted on.

Given,

A cube is made up of 27 smaller cubes.

The bigger cube is painted on the outside.

To Find,

No. of smaller cube which was painted on:

(a) 3 faces

(b) 2 faces

(c) 1 face

(d) no faces

Solution,

This problem could be solved using a simple method.

A painted cube has been made up of 27 identically sized smaller cubes and arranged in a 3 × 3 × 3 pattern.

The middle cube (of the three in each axis direction: x, y, and z) is completely unpainted.

⇒ 1 cube would have no faces painted on.

Now, the bigger cube has 8 corners. Each of these would be painted on 3 faces.

⇒ 8 cubes would have 3 faces painted on.

Further, a cube has 12 edge lines. Among the 12 edge lines, there are one cube each remaining which shares its faces with 2 faces of the bigger cube. Each of these would be pained on 2 faces.

⇒ 12 cubes would have 2 faces painted on.

Now we have 6 smaller cubes left. Those will be pained on 1 face.

⇒ 6 cubes would have 1 face painted on.

Hence, no. of smaller cube which was painted on:

(a) 3 faces = 8 cubes

(b) 2 faces = 12 cubes

(c) 1 face = 6 cubes

(d) no faces = 1 cube

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