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A cube of dimensions 8*8*8 is painted with red color. Now the cube is broken in smaller unit cubes.now find how many cubes are painted with red of i)one

side ii)two sides iii)three sides iv)no sides ? plz. fast

In how many smaller unit it is broken
how many cubes were broken


Visualize the large cube broken into smaller bits by making 8 uniform cuts along its length, breadth and height. There are some corner cubes, some edge cubes, some face cubes and some cubes on the interior. (see attachment).
The cubes which are painted on three sides must be the corner cubes which are 8 in number.
The cubes which are painted on two sides must be the cubes on the edges which are 12*6 = 72 in number.
The cubes which are painted on one side are the face cubes which are 6*6*6 = 216 in number.
The cubes which are not painted at all are the cubes in the interior which are 6*6*6 = 216 in number.
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Cube of dimensions :  8 * 8 * 8 . 
If it is divided into unit cubes.  we have  8^3 number of cubes = 512.

The unit cubes present at the pointed corners (vertices) of the big cube will be painted on 3 sides.  There are 8 corners to the cube.  So 8 unit cubes are painted on 3 sides.

The cubes present on the edges of cube, have two sides, exposed on two faces of the cube.   So they will be painted in two colors.   We have already taken care of the two unit cubes present on the corners.  So we will have  8-2 = 6  on each edge.  There are 12 edges.  Hence   72 cubes are painted on two dies.

The unit cubes that are on 2nd, 3rd, 4th, 5th, 6th, 7th row and 2nd 3rd, 4th, 5th and 6th, 7th column  will be exposed only one side.  So they are painted only on one side.  Hence,   6 X 6 = 36 unit cubes on each face.  There are 6 faces of a cube.  Hence, 36 *  6 = 216 unit cubes are painted on one side.

The number of cubes not painted on any side are 512 - 8 - 72 - 216 = 216 .  These are inside the big cube and not exposed to outside.  In fact the number can also be calculated as there are 6 * 6 * 6 in each dimension , hidden inside the big cube.

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