answer the following question?

Question

A cube is colored red on two opposite faces, blue on two adjacent faces and yellow on the two remaining faces. It is then cut into two halves along the plane parallel to the red faces. One piece is then cut into four equal cubes and the other into 32 equal cubes.How many cubes have atleast one blue face?

Answer 1

Assuming the interior of the cube is not blue, then 17.

Answer 2

17

Answer 3

Let's proceed step by step: first of all, to simplify the reasoning, try to visualize the cube as being put over a table, so that the red faces are the top and bottom faces. Then, the yellow and blue will be the vertical faces of the cube. As we know that the two blue faces are adjacent, then the two yellow faces (the remaining) also will be adjacent.

Now, imagine that we cut the cube at the middle, by the plane parallel to top-bottom red faces - then we obtain two half-cubes, each one with a big red face parallel to a non-colored face, and 4 half-sided faces yellow and blue (yellow adjacent, blue adjacent).

Now, consider the first half, that is divdided in 4 parts cubes: the 4 cubes obtained are as follows:
- all of them have a face red and an opposite face non-colored
- for the other 4 faces (the vertical ones), 2 are colored and 2 are non-colored (interior).

The 2 colored faces vertical are, for each of the 4 cubes, as follows: blue-blue, blue-yellow, yellow-yellow and yellow-blue

So we have that 3 of the 4 cubes have at least a blue face.

Now, let's consider the other half, that is divided in 32 small cubes. The process for such a division can be imagined as being in 2 steps, as follows:

step 1) we divide the original big half-cube in 4 small cubes, simlar as we discussed previously.

step 2) we divide each of these 4 small cubes in 8 smaller cubes, obtaining a total of 32 = 4*8 very small cubes. To divide a cube in 8 cubes is very easy: just cut the cube by the half and then subdivide each half in 4 cubes.

Now, we know (from what we discussed previously) that after step 1 are obtained 4 cubes as follows:

CUBE 1: has 1 red side (horizontal), 2 blue sides (vertical) and the rest if non-colored

CUBE 2 and 3: have 1 red side (horizontal), 1 blue side (vertical) and 1 yellow side (vertical)

CUBE 4: has 1 red side (horizontal), 2 yellow sides (vertical).

Now, we will subdivide each of these 4 CUBEs in 8 smaller cubes, and count how many of them have a blue side.

Let's start with CUBE 1:

To subdivide CUBE 1 in 8 smaller cubes we proceed in 2 steps, that I'll call A and B:

Step A) divide the cube in 2 half (by an horizontal plane parallel to the red plane), and then we obtain 2 half-cubes, each one with a big horizontal red side and 2 blue sides vertical (and adjacent).

Step B) subdivide each of these 2 half-cubes in 4 parts, and we obtain 6 cubes that have blue sides (3 cubes with blue sides at each half, as was already explained in the similar case we saw at the very first step of the explanation).

So, when subdividing CUBE 1 in 8 smaller cubes, 6 of them have a blue side.

Now, with CUBE 2 we do the same:

Step A) We obtain 2 half-cubes, each with a big horizontal red face and 1 vertical blue face

Step B) When subdividing each half in 4 only 2 of the 4 sub-cubes has a blue face (the other 2 have non-colored faces), and so only 4 (of the 8) subcubes total will have blue faces.

So, when subdividing CUBE 2 in 8 smaller cubes, 4 of them have a blue side.

Similarly, when subdivigind CUBE 3 in 8 smaller cubes, 4 of them have a blue side (CUBE 3 and CUBE 2 are similar).

Finally, as CUBE 4 had not blue sides, all the 8 cubes obtained from it will not have blue sides.

In conclusion, we obtained: 6 cubes with blue sides from CUBE 1, 4 cubes with blue sides from CUBE 2, other 4 from CUBE 3, and no one from CUBE 4. In conclusion: 14 (6+4+4) small cubes have blue sides (from a total of 32).

SO, WE FOUND THIS:

From the first half, 3 of the 4 cubes had blue sides.
From the second half, 14 of the 32 cubes had blue sides.

So, in total, 17 cubes will have blue sides.

Answer 4

no idea y would u wont ta kno dat anyway?

Answer 5

wtf? i cant even be bothered ATTEMTING to answer that question. Sorry mate, its the holidays...