No sooner was I done finishing a puzzle that I began to wonder the odds of a n-piece puzzle being solved on the first try by a spider-like mechanical assembler with n arms, each able to simultaneously place down one piece.
- All puzzle pieces have the same number of sides d.
- The number of possible orientations is constant amongst all pieces.
- In order to be 'solved', the pieces have to be in the correct order and in the correct orientation.
The probability that all pieces are in the correct order, I suppose, would be: $ \frac{1}{n!} $
The second component would be the odds that all the pieces are in the correct order. The odd of any one piece being in the correct orientation, I think, would be: $ \frac{1}{d} $
This is because a d-sided piece can be oriented in d different ways, assuming that all pieces have the same amount of possible orientations. (I get a gut feeling this is true, but cannot prove it.) Since there are n pieces, the odds of all pieces being oriented correctly would be: $\frac{1}{d^n}$
Combining both probabilities yields us the final odds: $ \frac{1}{d^nn!} $
Is this reasoning sound?