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You have N identical cubes in Euclidean space. Any cube can be fused face-to-face with any other cube -- at mo?
You have N identical cubes in Euclidean space. Any cube can be fused face-to-face with any other cube -- at most one cube per face.
All N cubes are attached to one another, directly or indirectly. For N=1, the only possible shape is a cube. For N=2, the only possible shape is a box one unit by one unit by two units.
How many distinct shapes are possible for N=3? N=4? Higher numbers of N? Can the solution to this problem be generalized to a formula for all positive integer Ns?
1 Answer
- ?Lv 48 years ago
Finding the value for N=1 through 4 is fairly easy. I don't think anyone has come up with a formula for the general case. If you check the reference, you will see that there is not a formula even in the two dimensional case of building shapes from squares.
For N =3, there are two possibilities. The three cubes can be arranged in a straight line or the third cube can be placed on a lateral side of one of the other two cubes to form a wedge shape.
For N=4, it starts to get complicated.
If three of the cubes are in a straight line then the fourth cube can be placed in three positions. It can be added so there are 4 in a row, it can be placed on a lateral side of one of the end cubes to form an L shape or it can be attached to the center cube to form a T shape.
If three of the cubes form a wedge shape then the 4 possibilities are:
Ignore the L shape because we have already done that.
Place the cube on one of the lateral sides of an end cube in three different ways: attach the cube to both end cubes, form a Z shape, form a twisted 3 dimensional Z shape.
There is one new way of attaching to the center cube (the other way is the T shape, which we have already done).
Total number of possibilities = 3 + 4 = 7.
