Anybody wants to count in real…

most of us have **3*3 Rubik’s cube** try it.

Exactly matches to this problem. 😊

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**Option C**

Total $27 (3 \times 3 \times 3)$ small cubes of $1$ unit each will be required to form a bigger cube of side $3$ units

No. of faces per cube $= 6$

Total number of cubes $= 9\times 3 = 27$

Total number of faces $= 27\times 6 = 162$

Total number of non visible faces $= 162-54 = 108$

$\frac{\text{No. of visible faces}}{\text{No. of non visible faces}}= \frac{54}{108}=\frac{1}{2}$

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We need a total of 27 small cubes to make one large cube.

1 small cube has a total of 6 faces, hence for 27 small cubes the total number of faces = 27 * 6 = 162

i.e One large cube has a total of 162 faces some of which are visible and some of which are not visible.

Now we will find the number of visible faces in 1 large cube.

From the figure, we notice that some pieces *(like corners) *have more faces visible, whereas some pieces *(like centre) *has only one face visible. Hence we first identify each type and count of visible faces for each type of piece

- Corner:

Now each face has 4 corner pieces. Hence 6 faces would have 6*4 = 24 corners pieces. But we have over-counted here. Notice that, each corner piece is shared among 3 faces, hence the correct number of corner pieces = 24/3 = 8 pieces.

Each corner piece has 3 faces visible, hence total visible faces for corner piece = 8*3 = 24 ----- (i)

- Side:

Each face has 4 side pieces. Hence 6 faces would have 6 * 4 = 24 side pieces, but again we have over-counted. Notice how each side piece is shared among 2 faces, hence the correct number of side pieces = 24/2 = 12 pieces.

Each side piece has 2 faces visible, hence total visible faces for side piece = 12 * 2 = 24 ----- (ii)

- Center:

Each face has 1 centre piece. Hence 6 faces would have 6 centre piece. But are we again overcounting? NO, the centre piece is never shared among any face of a large cube, hence no over-counting. So total centre piece = 6.

Each centre piece has 1 face visible, hence total visible faces for center piece = 6 ----- (iii)

Adding equation i, ii and iii we get 24 + 24 + 6 = 54 visible faces.

Now total invisible faces = Total faces – Total visible faces

= 162 – 54

= 108

Hence the proportion of the number of faces of the smaller cubes visible to those which are NOT visible

= 54 : 108

= 1 : 2