A CUBE OF 125 CMS HAS TWO ADJACENT SIDES PAINTED IN RED AND TWO ADJACENT SIDES PAINTED IN GREEN AND OTHER 2 ADJACENT SIDES IN BLUE
HOW MANY CUBES HAVE EXACTLY 2 SIDES PAINTED?
HOW MANY CUBES HAVE EXACTLY 1 SIDES PAINTED?
HOW MANY CUBES HAVE EXACTLY 0 SIDES PAINTED?
HOW MANY CUBES HAVE EXACTLY 2 SIDES PAINTED WITH 2 DIFFERENT COLORS?
The way this question generally goes is that the entire cube is cut into pieces to give smaller cubes of sides 1 cm each. Then you are asked these questions.
Take easy-to-visualize numbers.
Say, the side of the painted big cube is 3 cm. When you make small cubes of 1 cm side out of it, you can make 27 cubes out of it. (To verify, notice that the volume of the big cube is 3*3*3 = 27 cm3 so you can make 27 cubes of volume 1 cm3 out of it)
The cubes which were on the corners of the big cube will have 3 colored faces. The cubes which were on the edges of the big cube (but not on the corner) will have 2 faces painted. The cubes which were on the face of the big cube (but not on edges) will have one face painted. The cubes lying the middle i.e. hidden from view in the big cube will have no face painted.
A cube has 8 corners so 8 small cubes will have 3 faces painted.
Each edge of a 3*3*3 cube will produce 3 cubes 2 of which will lie on the corners. So one cube from the edge will have 2 faces painted. Since a cube has 12 edges, we will get 12 small cubes with 2 faces painted.
Each face of the cube gives 9 small cubes of which 8 will lie on the edges. 1 cube in the middle will have only 1 face painted. Since a cube has 6 faces, we will get 6 cubes of 1 face painted each.
Now visualize this: You slice one layer of cubes from each of the faces of the big cube to remove all color. Now the cube you are left with will make you small cubes which will have no face painted. In case the big cube has side 3 cm, you will shave off 1 cm from top face and 1 cm from bottom face. Similarly, you will shave off 1 cm from each of the other faces. You will be left with a single small cube with no color.
Now, to verify, notice that we get a total of 8 + 12 + 6 + 1 = 27 total cubes.
Try to use the same logic with larger cube size now.
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