Question 27

A cube is painted red on two adjacent faces and on one opposite sides face, yellow on two opposite faces and green on the remaining face. It is then cut into 64 equal cubes. How many cubes have one red coloured face only?

Solution

Cutting the cube into 64 equal pieces is cubes of $$4\times4\times4$$.

On any face, the number of small cubes that have exactly one painted face are the interior face-centre pieces (not on edges or corners) is equal to $$(n-2)^2$$ .

As we know from the given data that in a cube there are exactly 3 red painted faces in a cube, so the no of cubes which have one red coloured face only = $$3\times\ (n-2)^2\ =\ 3\left(4-2\right)^2=12$$ .

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