Mathematics question

Question

Daud want to paint some faces of a cube with green paint. At least one face must be painted. How many ways are there for him to paint the cube? Note: Two colorings are considered the same if one can be obtained from the other by rotation.

Answer 1

There may be a more advanced way to do this, but this problem is reasonably countable:

1 face: 1 way

2 faces: 2 ways (adjacent and opposite)

3 faces: 2 ways (3 around a corner, three in a line: start with 2 adjacent and there are two ways to put the third. Start with 2 opposite, there is one way, but it repeats the "in a line")

4 faces: 2 ways (symmetric with painting 2, you are not painting 2)

5 faces: 1 (sym. with painting 1)

6 faces: 1 way.

Answer 2

The wording of this question was a bit confusing. I hope this is the explanation you are looking for...

To solve this, I am going to look at ho wmany different ways the cube can be painted when 1, 2, 3, 4, 5, or 6 sides are painted. For the arrangements more complicated to explain, I will also refer to the numbers on a standard six-sided die.

1 side painted: 1 way - 1 side painted

2 sides painted: 2 ways - opposite sides painted (3 and 4 on a die), 2 adjacent sides painted (4 and 2 on a die)

3 sides painted: 2 ways - 3 in a row forming a horseshoe (1, 3, and 6 on a die), 3 that all share one vertex (1, 4, and 5 on a die)

4 sides painted: 2 ways - opposite sides unpainted (1, 2, 5, and 6 on a die), 2 adjacent sides unpainted (1, 3, 5, and 6 on a die)

5 sides painted: 1 way - 1 side unpainted

6 sides painted: 1 way - all sides painted

So, in total, there are 9 ways that the cube can be painted.

Answer 3

There is 1 way to paint 1 face of the cube

There are 2 ways to paint 2 faces of the cube (adjacent faces or opposite faces)

There are 2 ways to paint 3 faces of the cube (no opposite faces and one pair of opposite faces plus a face adjacent to both)

There are 2 ways to paint 4 faces of the cube (same as painting 2 faces)

There is 1 way to paint 5 faces of the cube (same as painting 1 face)

There is 1 way of painting all 6 faces of the cube.

Tallying everything up

1 + 2 + 2 + 2 + 1 + 1 = 9 different ways to paint the cube