hidden faces of cubes. i have to come up with a formula which can tell thenumber of hidden faces of cube?

Question

by looking at the shown faces

please help asap! im really stuck on this one

Answer 1

If you had just one cube (1 x 1 x 1), there would be 6 external faces and 0 internal faces.

If you had eight SMALLER cubes (2 x 2 x 2), there would be 4 external faces on each side of the BIGGER cube, so 6 x 4 = 24 external faces. Each SMALLER cube has 6 faces, and there are 8 cubes, so there are 48 faces total. 48 total faces - 24 external faces = 24 internal faces.

If you had 27 SMALLER cubes (3 x 3 x 3), there would be 9 external faces on each side of the BIGGER cube, so 6 x 9 = 54 external faces. Each SMALLER cube has 6 faces, and there are 27 cubes, so there are 162 faces total. 162 total faces - 54 external faces = 108 internal faces.

Do you see the pattern?

If n is the number of cubes along a single edge of the BIGGER cube, then each BIGGER cube is composed of n × n × n = n³ cubes. On each side of the BIGGER cube, you would have n × n = n² external faces. Every BIGGER cube has 6 sides, so you would have 6n² external faces.

But you have n³ SMALLER cubes, and each of THEM has a total of 6 faces, for an overall total of 6n³ faces. So the number of internal faces is:

# of total faces - # of external faces = # of internal faces

6n³ - 6n² = 6n²(n - 1)

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There's a very good article on this topic on the mathforum website.

http://mathforum.org/library/drmath/view/56799.html