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Count the number of distinct color patterns of the wristband?
The inside and outside of the wristband has the grid with 3 rows and 6 columns. (we call the direction of the hole of the wristband "column.") And we have 5 colors to color each box. How many distinct color patterns can we make?
Additional rules:
-The wristband can freely be rotated in any directions. And if 1 pattern looks the same as another when rotated, they are counted as 1 pattern.
-The wristband can be turned inside out. And if 1 pattern looks the same as another when turned inside out, they are counted as 1 pattern.
-The wristband can be rotated and then turned inside out, or vice versa. And if 2 patterns look the same after rotating and turning inside out, they are counted as 1 pattern.
Scythian,
In our case, we have 36 boxes to color.
Scythian,
I just learned Polya's enumeration theorem, and wondered how to use it in this sort of situation. So, we cannot use it in this situation?
How to use Polya's enumeration theorem:
http://arxiv.org/ftp/arxiv/papers/1001/1001.0072.p...
More examples + proof:
1 Answer
- Scythian1950Lv 71 decade agoFavorite Answer
Do we have 18 or 36 boxes to color? That is, sometimes a wristband is made so that it has the same color on both sides at any given point.
Edit: I think a book can be written on just this one problem.