How many ways is it possible to color the faces of a six sided black cube white?

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

arrangements in which all six faces are the same color and
arrangements in which exactly five faces have the same color. In each of these cases the cube can be placed so that the four vertical faces have the same color. The only other suitable arrangements have four faces of one color, with the other color on a pair of opposing faces. Since there are three pairs of opposing faces, there aresuch arrangements. The total number of suitable arrangements is therefore, and the probability is

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

. We need to compute.

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.

How many ways can you color a cube with 6 colors?

Hence, the answer is 5*6 = 30 ways.

How many different ways can the cube be painted in black and white?

Hence, the cube can be painted in 10 different ways.

What are the number of ways to paint the faces of a cube with 6 different colors?

Hence, the correct answer is 30 Ways.

How many ways can you paint a cube?

There are 57 distinct ways to paint a cube with three colours. The cube has 6 faces. If we were allowed to count non-rotationally distinct ways of painting the cube as different ways the problem would be easy, as the hint suggests: 3^6 = 729.

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