Trial and Error
I recently got three metal screen doors powder coated. I just had to reassemble them.
Each screen door had five similar pieces of metal holding down the wire mesh. Each piece of metal can only go in one way around, but there are four orientations that may need to be tried (turning over, and swapping end-to-end).
By my calculations there are 60 possible choices for placement of the first piece of metal (15 pieces x 4 orientations) and 56 for the second and so on. This gives the total possible combinations to be:
N = 60 x 56 x 52 x … x 8 x 4
= 1404104661094367232000
I started to get worried. Then I realised that on average, the first piece should take around 30.5 tries (8 pieces on average; the first 7 need to try 4 orientations, the last takes 2.5 on average), the second piece around 28.5 tries and so on. So the number of tries should be:
N = 30.5 + 28.5 + 24.5 + … + 2.5
= 247.5
Much better…
It actually only took me about 10 minutes to sort them out. There were just enough differences between some of the pieces that some of the choices were significantly reduced. By doing it in the right order, I think I only needed a hundred or so tries altogether.
I think I actually spent longer doing the calculations and blogging about it than actually doing the job 
December 18th, 2009 at 7:16 am
Actually if all the pieces were literally indistinguishable you would only have to attend to the orientations. If there were distinguishable then the differences would be matched in the pockets in the door and would resolve the possible ambiguity. Once again you would only have to attend to the orientations.
December 18th, 2009 at 10:41 am
You are correct.
The problem was that the pieces were nearly the same, but not quite. If they were identical, there would be no problem. If they were very different, it would be easy too.