From the 1970s I have a Hexomino called
Computer Puzzle in Computer Age
promising more than 1.000.000 solutions. A later version with a different title (I could not remember) says more than 1.000.000.000 solutions are possible. Probably the number is closer to 1.000.000.000.000 - but of course a single solution is hard to find by hand.
In this Hexomino the 35 pieces are arranged in a rectangle 19 x 11 plus a nose in the middle of one of the longer sides. (This has a specific reason, because it is not possible to arrange the 35 pieces in a square.)
Especially for this arrangement I'd like to find all solutions.
The puzzle looks like this (could not put into comment):
┌─┬───┬─────┬─────────┬───┬─┬─────┬───┐
│ ├─┐ └───┐ └───┐ ┌─┬─┘ │ │ ┌─┤ │
│ │ └─┐ ┌─┴───┬─┴─┘ ├─┐ ┌─┤ ├─┐ │ │ │
│ └─┐ ├─┤ ┌─┐ │ ┌─┐ │ └─┘ │ │ └─┤ └───┤
│ ┌─┘ │ │ │ └─┼─┘ └─┴─┐ ┌─┤ ├─┐ └───┐ │
├─┼───┘ └─┤ │ ┌─┬───┴─┘ │ │ └─┬───┴─┤
│ └─┬───┬─┴─┐ ├─┘ └───┬─┐ ├─┴─┐ └─┐ │
│ │ ┌─┴─┐ ├─┴─┐ ┌─┬─┘ └─┤ ├─┐ ├─┐ │
│ ┌─┘ │ │ │ ┌─┴─┘ └─┬─┐ │ ┌─┘ └─┘ ├─┤
├─┤ ┌─┘ ┌─┘ │ ├─┐ ┌───┘ │ │ ├───────┤ │
│ └─┴───┴─┬─┘ │ └─┴───┐ └─┼─┘ ┌─────┘ │
└─────────┴───┴───┐ ┌─┴───┴───┴───────┘
└─┘