How many unique shapes can you get by unfolding a paper cube?
You can only cut along edges, and the shape must be in one piece and flat. By unique, I mean rotations and reflections don't count.
This problem can be analogized to four dimensions as well. How many unique 3dimensional shapes can be made by "unfolding" a 4dimensional hypercube into 8 cubes? This problem is significantly more difficult than the first.
In finding all the possibilities for the puzzle of part 1, I found the best way was to start with two adjacent squares and then build by adding squares in every possible adjacent location except if that configuration was already produced by some other path. I use *'s below to represent the squares.
So we start with
**
We can add another in two places:
*** **
*
We can add another to the first one in three places, but then can add another to the second in only one place that both allows folding into a cube and also doesn't repeat a pattern from the first:
**** *** ***  **
* * **
Note that here, and below, a vertical bar separates those derived from different configurations of the generation above.
The fifth square can be attached as follows:
* * * * *
**** ****  *** *** ***  ***  *** **
* * * * * * ** **
And the sixth as follows:
*
* * * * * * ** * * **
**** **** **** ****  **** ****  ***  *** ***    ***  **
* * * * * * * * ** *** **
Note that in the final instance, a couple of configurations from the preceding row did not produce any new configurations, because they would repeat previous configuations on the new row.
There are 11 configurations on the last row.
Edited on July 22, 2004, 3:28 pm

Posted by Charlie
on 20040722 15:24:54 