cube sum again- with painted faces?

Question

A cube of side 12 cm is painted red on all the faces and then cut into smaller cubes, each of side 3 cm. What is the total number of smaller cubes having none of their faces painted?
[1] 16 [2] 8 [3] 12 [4] 24

similarly how can find the number of cubes with 1 face painted and 0 face painted etc/?....?? please explain,,, give me the general formula........

Answer 1

The big cube is cut into 4 x 4 x 4 = 64 small cubes

There are 8 original corners - 3 painted sides
There were twelve edges on the original cube - each will give two cubes with two painted sides - 24 in all
There were 6 faces - each with four cubes with one painted side - 24 in all

So the number with no painted sides is 64 - 8 - 24 -24 = 8

(The unpainted cubes formed a small cube 2 x 2 x 2).

Answer 2

If a cubic block of n*n*n individual cubes is taken, and the outer layer of cubes is removed (which is essentially what you are doing in this question), you will be left with a block of (n-2)*(n-2)*(n-2) cubes.

There will always be 8 cubes with 3 faces painted (the corner cubes)

A cube has 12 edges
Each edge will have (n-2) cubes that are not corner cubes
There will be 12(n-2) cubes that have 2 faces painted.

A cube has 6 faces
Each face will have (n-2)^2 cubes that have only one face painted.
There will be 6(n-2)^2 cubes that have only one face painted

Unpainted cubes: (n-2)^3
One face painted: 6(n-2)^2
2 faces painted: 12(n-2)
3 faces painted: 8

(n-2)^3 + 6(n-2)^2 + 12(n-2) + 8
= n^3 - 6n^2 + 12n - 8 + 6n^2 - 24n + 24 + 12n - 24 + 8
= n^3 cubes in total

In this question n=4
Unpainted cubes: 2^3 = 8
One face painted: 6*2^2 = 24
2 faces painted: 12*2 = 24
3 faces painted: 8

Answer 3

The number of unpainted cubes is 2^3 = 8.

Since there are 4 cubes per side with 1 painted face therefore there are 4 * 6 = 24..

There are 2 cubes per edge with 2 painted faces therefore there are 2 * 12 = 24.

There is 1 cuber per corner with 3 painted faces therefore there are 1 * 8 = 8.

Answer 4

Your cube is cut into 4 cubes to a side. All of the outside cubes have at least one face painted. Take them away and you have a stack 2X2X2 on the inside totally unpainted, or 8 unpainted cubes.

For one side consider that each cube will have at least one side painted. The ones on the end will have two sides painted. That leaves you the 4 in the middle with one side painted. 4 X 6 sides is 24 cubes with one side painted.

Don't know about a general formula. Sometimes you just have to think.

Answer 5

ok my asian brother. 2. A triangle with a community of a hundred and twenty mm2 has the peak of 10mm. Wha tis the part of a similar triangle with a height of 20mm? part of triangle: A =a million/2 x h x b section = a million/2 x height base 120mm² = a million/2 x 10 x base discover out the backside: 120mm² = 5 x base base = a hundred and twenty/5 base = 24mm. A triangle this is comparable could have comparable boost in height and base dimensions. seeing that height is doubled, then base is doubled. height is 20mm base is 48mm section = a million/2 x 20 x forty 8 = 480mm² 3. each and each face if a 5x5x5 cube is painted purple. This cube is then cut back into one hundred twenty five unit cubes. how most of the unit cubes have not any faces that are painted purple? 5 x 5 x 5 = one hundred twenty five this suggests that the purple painted cube has been cut back into one hundred twenty five cubes of a million x a million x a million instruments. On one face of the 5 x 5 x 5 purple painted cube there is 25 faces of a million x a million squares from the smaller cubes. there is six faces of the vast purple painted cube so there could be 25 x 6 faces = a hundred and fifty faces of the smaller cubes which have been painted purple. Now we could discover the comprehensive quantity of faces of each and every of the little cubes cut back out. each and each small cube has six faces and there are one hundred twenty five of them. 6 x one hundred twenty five = 750 faces. So the faces which at the instant are not painted purple are 750 - a hundred and fifty = six hundred faces.