solid geometry cube

[francoisph]

A cube of side 5cm is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?

someone could explain with pic please
I didnt get this issue at all


thks

[asamaverick]

Look at the square above, assume it's a square with sides as 5 cm. I have divided this into 25 blocks (each a square with side as 1cm). This is basically one face of the cube in your question. There will be 6 faces like these in the cube and the only ones with just a single side painted are the ones shaded in brown. All the others will have at least 2 faces painted.

So there are 9 of them on each face of the cube, hence 9*6 = 54 is the answer you are looking for.

[kvcpk]

Is the OA 54?

[Rich@VeritasPrep]

The volume of this cube is 5*5*5 = 125 cm^3. So if you slice the cube up into smaller cubes that each have volume 1 cm^2, you end up with 125 smaller 1x1 cubes.

Each of the six faces of the cube has a surface area of 5*5 = 25 cm^2. This consists of 25 smaller squares, each with area 1 cm^2 (i.e. 1 cm by 1 cm).

Looking just at one of the faces, all of the smaller 1x1 squares on the perimeter of the larger 5x5 square (denoted by 'x' in the below diagram) will be part of a 1x1x1 cube that is either in the corner of the large cube or along one of the larger cube's edges:

x x x x x
x o o o x
x o o o x
x o o o x
x x x x x

Each of the x's denotes part of a 1x1x1 cube that will have more than one face painted, and thus we are not interested it.

Interestingly enough, notice that the o's form a 3x3 square for a total of 9. Each of these 9 is part of a 1x1x1 cube that has no other face painted (since the other faces came from inside the cube).

There are 9 such cubes gotten from each of the 6 faces. So the answer is 54.