# A lettered cube has 6 sides with a different letter on each side. Each letter can be shown in 4 different positions. How many different ways?

### 7 Answers

- roderick_youngLv 711 months ago
Are you asking how many different ways could the cube come up? That exactly describes the situation for a cube in the game Boggle. A cube has 6 different letters, and can fall into the matrix in any one of 4 orientations. Excellent answers from @rotchm and @Puzzling already.

Source(s): I was the author of a Boggle-type program back in the '80's. - atsuoLv 612 months ago
First , assume that each letter must be shown on the center of its side ,

and the direction of a letter is disregarded .

Let the 6 letter be A,B,C,D,E and F .

Let type1 cube be the cube such that A and B are shown on opposite sides ,

and put type1 cube so that it becomes top side is A and bottom side is B .

4 letters C,D,E,F can be shown in other 4 sides with any combination ,

but we can rotate the cube around the vertical axis so 4 cubes become the same cube .

Therefore 4! / 4 = 6 different type1 cubes exist .

Let type2 cube be the cube such that A and B are shown on adjacent sides ,

and put type2 cube so that it becomes top side is A and front side is B .

4 letters C,D,E,F can be shown in other 4 sides with any combination ,

and we can not rotate the cube because it must be that top side is A and front side is B .

Therefore 4! = 24 different type2 cubes exist .

So 6(type1) + 24(type2) = 30 different cubes exist .

Next , think about "Each letter can be shown in 4 different positions" .

We can move each letter from the center to 1 of 4 different positions ,

so 1 cube can be modified into 4^6 = 4096 cubes .

Therefore total of 30 * 4096 = 122,880 cubes (ways) exist .

If "4 different positions" means "4 different directions" then the result is unchanged

because we do not move each letter but we can rotate each letter .

(I think the direction of a letter must be parallel to an edge . If the direction of a letter

may not be parallel to an edge then the problem becomes complicated .)

- NickLv 612 months ago
Assuming you have 6 different letters each with no rotational symmetry e.g. A,B,C,D,E,F.

If this is a question about a *fixed cube* then each face is *distinct* to begin with and there are 6! ways to place letters on faces and 4^6 ways for them to be oriented: 6!*4^6 ways.

or

If this is a question about a *free cube* (i.e. equivalent under the symmetry group of rotations) then place A on a face to define it as the bottom face, let the orientation of this letter fix front, back, left and right faces, this then means that we may place the remaining 5 letters on 5 remaining *fixed* faces in 5! ways and in 4^5 orientations: 5!*4^5 ways.

Notice the second count is 1/24 of the first. This is because the assumed equivalence under rotation reduces the possible configurations.

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- Φ² = Φ+1Lv 712 months ago
Can you please confirm:

1) How many letters there are from which the six are chosen?

2) Can you please complete the question "How many different ways?"? I suspect the two good answers you already have have been "thumbed down" because they (and I) are having to make assumptions which may be incorrect.

________8>< _ _ _ _ _ _ _

I started this, so I should give an answer. The final answer has been given by others, put the process is spelled out simply.

Pick a cube and place the first letter on the front face in "standard" orientation. This is our starting point.

Any one of the five remaining letters can be placed on the opposite face (back) in any one of four orientations. (20)

Any one of the four remaining letters can be placed on the top face in any one of four orientations. (16)

Any one of the three remaining letters can be placed on the right face in any one of four orientations. (12)

Any one of the two remaining letters can be placed on the left face in any one of four orientations. (8)

The remaining letter can be placed on the bottom face of the cube in any one of four orientations. (4)

This produces a total of 20 × 16 × 12 × 8 × 4 = 122,880 unique ways the six letters can be laid out on the cube.

- PuzzlingLv 712 months ago
This is known as the fundamental (or basic) counting principle:

"When there are m ways to do one thing, and n ways to do another, then there are m × n ways of doing both."

Read more at the link below.

There are 6 choices for which letter is facing up.

There are 4 orientations for that letter.

So, what's your conclusion?

- rotchmLv 712 months ago
For the first letter, there are 4 ways to display it.

For the second letter, there are 4 ways to display it.

Etc...

For the sixth letter there are 4 ways to display it.

Making a grand total of [left for you].

Done!