By sheer chance when solving a hexomino thing a few days ago I was presented with a really nice clear example of a technique that I'm sure I always just referred to in blog posts and things as 'piece substitution' but never actually ever bothered to clarify. So here it is. Consider yonder picture:
The two pieces in red to the side are the two I'm left with, and the hole remaining just won't accommodate them in any way short of physically snapping the pieces apart. The best we can do is getting the more irregular piece in there in the obvious place, leaving a longer thinner 'L' shaped gap than we're capable of filling. Like this.
The trick here is to look at the two pieces in light blue, one of which is the long skinny 'L' piece we need. Notice that we can do this:
which uses up our unusable P-shaped piece, and at the same time frees up out long piece, allowing us to fill the other hole and complete the puzzle.
Of course, there's no guarantee that it'll fall into place as nicely as that. Sometimes it's two pairs of pieces that can make the same shape that need to be swapped, or sometimes it's even uglier, like a chain of substitutions that free up one particular piece then use that piece to free up another. But it's a viable technique surprisingly often given how much of an utter fluke it looks.
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