The document discusses problems involving cubes that are painted different colors on their faces and then cut into smaller identical cubes. It provides the answers to questions about counting the number of smaller cubes with a given number of faces of each color. For a cube cut into 216 smaller cubes where 3 pairs of adjacent faces are each painted a single color, there are 72 cubes with no red paint, 44 with at least two colors, 72 with only red or blue, 14 with only red and blue, and 18 with red and blue. For a cube cut into 343 smaller cubes where 3 faces are each a different color, there are 43 painted cubes total, 1 with 3 faces painted, 18 with 2 faces painted, 75 with 1 face painted, and

1. Problems on Cubes

2. 1. 64 smaller identical cubes are arranged to form a larger cube. How many
such similar type of smaller identical cubes are required to completely
cover the larger cube
Number of smaller cubes with 3 faces open ?
Number of smaller cubes with 2 faces open ?
Number of smaller cubes with 1 faces open?
Number of smaller cubes with 0 faces open?
2. A cube is painted on all its faces with red color. The cube is now cut into
343 smaller identical cubes?
Number of smaller cubes with 3 faces open ?
Number of smaller cubes with 2 faces open ?
Number of smaller cubes with 1 faces open?
Number of smaller cubes with 0 faces open?
3. A cube is painted on 3 pairs of opposite faces with same color red, blue ,
green . The cube is now cut into 216 smaller identical cubes?
Number of smaller cubes with no red paint?
Number of smaller cubes at least two different colors?
Number of smaller cubes only red or only blue?
Number of smaller cubes only red and blue?
Number of smaller cubes with red and blue?

3. • A cube has 6 faces, 12 edges and 8 corners
• Number of smaller cubes with 3 faces open = corners = 8
• Number of smaller cubes with 2 faces open = 12 (n - 2)
• Number of smaller cubes with 1 faces open = 6
• Number of smaller cubes with 0 faces open =

4. 64 smaller identical cubes are arranged to form a larger cube.
Number of smaller cubes with 3 faces open ?
Number of smaller cubes with 2 faces open ?
Number of smaller cubes with 1 faces open?
Number of smaller cubes with 0 faces open?
1.
Ans:
=> n = 4
Number of smaller cubes with 3 faces open = 8
Number of smaller cubes with 2 faces open = 12 (4 - 2) = 24
Number of smaller cubes with 1 faces open = 6
Number of smaller cubes with 0 faces open = = 8

5. A cube is painted on all its faces with red color. The cube is now cut into 343
smaller identical cubes?
Number of smaller cubes with 3 faces open ?
Number of smaller cubes with 2 faces open ?
Number of smaller cubes with 1 faces open?
Number of smaller cubes with 0 faces open?
2.
Ans:
=> n = 7
Number of smaller cubes with 3 faces open = 8
Number of smaller cubes with 2 faces open = 12 (7 - 2) = 60
Number of smaller cubes with 1 faces open = 6 150
Number of smaller cubes with 0 faces open = = 125

6. A cube is painted on 3 pairs of opposite faces with same color red, blue ,
green . The cube is now cut into 216 smaller identical cubes?
Number of smaller cubes with no red paint?
Number of smaller cubes at least two different colors?
Number of smaller cubes only red or only blue?
Number of smaller cubes only red and blue?
Number of smaller cubes with red and blue?
3.
Ans :
n = 6
Number of smaller cubes with no red paint = 144
Number of smaller cubes at least two different colors = 56
Number of smaller cubes only red or only blue = 64
Number of smaller cubes only red and blue = 16
Number of smaller cubes with red and blue = 24

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Problems for practice are given in the description of the video.

8. Practice problems
on Cubes

9. 1. A cube is painted on 3 pairs of adjacent faces with same color red, blue,
green. This cube is now cut into 216 smaller identical cubes
Number of smaller cubes with no red paint?
Number of smaller cubes at least two different colors?
Number of smaller cubes only red or only blue?
Number of smaller cubes only red and blue?
Number of smaller cubes with red and blue?
2. A cube is painted on 3 faces with 3 different colors red, green and blue.
This cube is cut into 343 identical cubes.
Number of smaller cubes painted?
Number of smaller cubes with 3 faces painted ?
Number of smaller cubes with 2 faces painted?
Number of smaller cubes with 1 faces painted?
Number of smaller cubes with 0 faces painted?

10. 1. A cube is painted on 3 pairs of adjacent faces with same color red, blue,
green. This cube is now cut into 216 smaller identical cubes
Number of smaller cubes with no red paint?
Number of smaller cubes at least two different colors?
Number of smaller cubes only red or only blue?
Number of smaller cubes only red and blue?
Number of smaller cubes with red and blue?
Ans :
n = 6
Number of smaller cubes with no red paint = 72
Number of smaller cubes at least two different colors = 44
Number of smaller cubes only red or only blue = 72
Number of smaller cubes only red and blue = 14
Number of smaller cubes with red and blue = 18

11. 2. A cube is painted on 3 faces with 3 different colors red, green and blue.
This cube is cut into 343 identical cubes.
Number of smaller cubes painted?
Number of smaller cubes with 3 faces painted ?
Number of smaller cubes with 2 faces painted?
Number of smaller cubes with 1 faces painted?
Number of smaller cubes with 0 faces painted?
Ans:
 n = 7
Number of smaller cubes painted = 43
Number of smaller cubes with 3 faces open = 1
Number of smaller cubes with 2 faces open = 18
Number of smaller cubes with 1 faces open = 75
Number of smaller cubes with 0 faces open = 300

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