Sam Loyd's unsolvable $14-15$ puzzle is well known to everyone. I also found it quite fascinating when I was playing with it in my childhood but didn't know about that one particular arrangement that cannot be solved. Now I do. But there is another question on my mind-
How many arrangements are there such that the puzzle in those arrangements cannot be solved? Can we calculate that number?
Half of all arrangements can be reduced to the solvable 14-15 arrangement, the other half is unsolvable. (Groups of 3 can be solved, or reduced to this unsolvable variation.)
More or less similar to Rubik's cube. Every solution can be reduced to one were one middle piece can be flipped or one corner piece is in one of 3 rotations, with 2 invalids. So for the cube, 5 out of 6 configurations are invalid.