Question

A cube has one of its sides equal to 20 cm. It is fully painted on all the surfaces. It is cut completely through the block to form cubes 1x1x1 cm.

Determine the number of cubes with only 2 faces painted.

Answer 1

side length of cube = 20

Total number of small cubes of 1x1x1 cm = S³ = 20³ = 8000 (where S is the side length of cube)

Number of small cubes having three faces painted: The small cube having three faces coloured are situated at the corners of the big cube because at these corners only three faces of the big cube meet. Therefore the required number of such cubes is always 8, because there are 8 corners.

Lets consider the diagram below:

ܬ ܚܬܐ ܙܬܬܠܛܝ ----ܐܝܝܐ

Number of small cubes having only two faces painted: From the above figure each edge of the big cube 4 small cubes are connected and two out of them are situated at the corners of the big cube which have all the three faces painted. So,to each edge two small cubes are left which have two faces painted. As the total no. of edges in a cube are 12, hence the number of small cubes with two faces coloured

Number of small cubes with two faces coloured =

Number of small cubes with two faces coloured = 18 * 12 = 216

where $x=\frac{Side \ of \ big \ cube}{Side \ of \ small \ cube}$

Answer:

number of cubes with only 2 faces painted. = 216