Question

After the surface area of a cube is painted, the cube is cut into 64 smaller cubes of same dimensions as in fig above. How many have no face painted? 1 face painted? 2 faces painted? 3 faces painted?
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Answer 1

⠀⠀ ⠀ ⠀⠀ $\huge{\tt{\red{❥}\green{A}\purple{N}\pink{S}\blue{W}\orange{E}\red{R}}}$

A cube is painted blue on all faces and it is cut into 64 small cubes of equal size. How many cubes are not painted?

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35 Answers



Abhinna Agarwal, studied at Delhi Public School

Answered November 1, 2017 · Upvoted by Dwight House, Ph. D. Mathematics, Duke University (1972)



Let this be the original cube .

It is cut into 64 small cubes ( demarcated by lines ) .

Let's go layer wise ( top to bottom )

There are 4 layers .

1st layer :-

All the cubes are painted .

No. Of unpainted cubes = 0

2nd layer :-

Only side cubes are painted , internal 4 cubes ( middle most ) are not painted .

No. Of unpainted cubes = 4

3rd layer:-

Same as layer 2

No. Of unpainted cubes= 4

4th layer:-

Same as layer 1 .

No unpainted cubes.

Total unpainted cubes= 8

Hope this answer helps .

Answer 2

Answer:

no face painted = 8

1 face painted = 24

2 faces painted = 24

3 faces painted = 8

Step-by-step explanation:

no face painted = 8 -A cube has 6 faces.lt is coloured on all faces. To cut it into 64 equal cubes, it is cut three times across the length and breadth.There are 16 cubes on each face(4 by 4). Now, if we remove the cubes on all sides, there will be 8 cubes left in the middle that are not painted.

1 face painted = 24- we can clearly see that the cubes that would have only one side painted would be the middle 4 cubes of all the faces, so, by adding up the middle 4 cubes of all the faces, we get 24.

2 faces painted = 24- here we observe that the middle 2 cubes at the top and bottom row and the middle 2 cubes in the middle of the first and the last columns of each face have 2 faces painted and when adding up while keeping in mind that we dont count the same cube twice in the faces, the result is 24.

3 faces painted = 8- the corner cubes are the ones that have 3 faces painted and there are 8 corners in this cube.So, there are 8 cubes that have 3 faces painted.