Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
Solution 1
[Alcumus solution]
If the orientation of the cube is fixed, there arepossible arrangements of colors on the faces. There are
Solution 2
Label the six sides of the cube by numberstoas on a classic dice. Then the "four vertical faces" can be:,, or.
Letbe the set of colorings whereare all of the same color, similarly letandbe the sets of good colorings for the other two sets of faces.
There arepossible colorings, and there aregood colorings. Thus the result is
Using the Principle of Inclusion-Exclusion we can write
Clearly, as we have two possibilities for the common color of the four vertical faces, and two possibilities for each of the horizontal faces.
What is? The facesmust have the same color, and at the same time facesmust have the same color. It turns out thatthe set containing just the two cubes where all six faces have the same color.
Therefore, and the result is
Suppose we break the situation into cases that contain four vertical faces of the same color:
I. Two opposite sides of same color: There are 3 ways to choose the two sides, and then two colors possible, so.
II. One face different from all the others: There are 6 ways to choose this face, and 2 colors, so.
III. All faces are the same: There are 2 colors, and so two ways for all faces to be the same.
Adding them up, we have a total ofways to have four vertical faces the same color. There areways to color the cube, so the answer is
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.