Question

343 cubes of similar size are arranged in the form of a bigger cube and kept at the corner of a room all the exposed surface are painted then how many cubes have at least two faces painted?

Answer 1

343 cubes of similar size are arranged in the form of a bigger cube (7 cubes on each side, i.e., 7 x 7 x 7) and kept at the corner of a room, all the exposed surfaces are painted then:

How many of the cubes have 0 faces painted?None of these

Correct Option: D

Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18 
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 1- 18 - 108 = 216
In other words number of cubes with 0 painted is (7 - 1)3 = 216.
From the above explanation number of the cubes with 0 faces painted is 216.

Answer 2

Answer:

The number of the cubes with at least 2 faces painted exists at $18 + 1 = 19.$

Explanation:

Given:

343 cubes of similar size are arranged in the form of a bigger cube and kept in the corner of a room all the exposed surfaces are painted.

To find:

How many cubes have at least two faces painted.

Step 1

Out of 6 faces, 3 faces are revealed and those stood painted.

The number of vertices with three faces exposed (Painted) exists at 1.

The number of vertices with 2 faces exposed (Painted) exists at 3.

The number of vertices with 1 face exposed (Painted) exists at 3.

The number of vertices with 0 faces exposed (Painted) exists at 1.

The number of sides with 2 sides exposed (Painted) exists at 3.

The number of sides with 1 side exposed (Painted) exists at 6.

The number of sides with no sides exposed (Painted) exists at 3.

Step 2

From the above observation,

The number of cubes with 3 faces Painted exists at 1.

A number of cubes with 2 faces Painted exists given by sides which are exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required the number of cubes exists $6 *3 = 18$.

The number of cubes with 1 face Painted exists given by faces that are exposed from one side and there are 3 such faces hence required the number of cubes exists $36 * 3 = 108$.

The number of cubes with 0 faces Painted exists provided by the difference between the total number of cubes - the number of cubes with at least 1 face painted $= 343 - 1- 18 - 108 = 216$.

In other terms number of cubes with 0 painted exists $(7 - 1)3 = 216.$

From the overhead description number of the cubes with at least 2 faces painted exists $18 + 1 = 19.$

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