Red-Blue Blocks
Probability
Level
3
A toy factory creates cube blocks in which each side is randomly colored red or blue. How many distinct cubes can they create?
Cubes are distinct if they cannot be obtained via rotations.
32
20
10
64
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1 solution
We consider cases based on the number of faces that are red
Case 1: 0 red
Clearly, just 1 cube.
Case 2: 1 red
Clearly, just 1 cube.
Case 3: 2 red
The red faces can have a common edge, or be on opposite sides.
There are 2 cubes.
Case 4: 3 red
The red faces could have a common vertex, of form a "straight line".
There are 2 cubes
Case 5: 4 red
This is identical to 2 red. 2 cubes
Case 6: 5 red
This is identical to 1 red. 1 cubes
Case 7: 6 red
This is identical to 0 red. 1 cubes
Hence, there is a total of 10 such cubes.