Cube question?

Question

cublets form a cube. Within that cube, these cublets also form several rectangular prisms.
How many of these rectangular prisms are non-cubical?

That means
1x1x2
1x2x2
1x2x3
1x3x3
2x2x3
2x3x3

I think that's them all.

Answer 1

First of all, I'm assuming, since the dimensions you gave for the rectangles don't go above three units in any direction, that the cube is 3x3x3.

Picture a slice of the cube which is 1x1x3. Then you can have two 1x1x2 rectangles within it facing in the same direction. There are nine of these slices in the cube which point in that direction, and 18 of the 1x1x2 prisms. There are three equivalent directions to point in, so there are a total of (3x9 =) 27 1x1x3 prisms and (3x18 =) 54 1x1x2 prisms. (Incidentally, you forgot to mention 1x1x3 in your list.)

Now picture a slice of the cube which is 1x3x3. Within this slice there are four 1x2x2 slices and four 1x2x3. Three large slices make up a cube, so in any given direction, there are three 1x3x3 and twelve each of 1x2x2 and 1x2x3. Again, the slices can go in any direction, so there are a total of nine 1x3x3 rectangles and 36 each of 1x2x2 and 1x2x3.

For 2x2x3, they can point in any of the three directions and go all the way through the cube in that direction. in the plane of the other two directions, their cross section is a 2x2 square, which can have four positions within a 3x3 square. So there's four of these in each direction, or 12 total.

For 2x3x3, you're basically just removing a face of the cube. There are six faces, therefore six 2x3x3 rectangles.

To recap:

1x1x2: 54
1x1x3: 27
1x2x2: 36
1x2x3: 36
1x3x3: 09
2x2x3: 12
2x3x3: 06

Total: 180

Answer 2

Sorry, I didn't get the question. I was gonna ask Game or Ice?

Answer 3

none of them, cuz squares can be rectangles, but rectangles cant be squares. i know, im still in 2d but, hey just givin it a shot. or is it all of em?

Answer 4

um wtvr