Oh nice. I don't actually know how to play solitaire, but I know Microsoft used the method of randomly generating them, solving them and then saving solvable seeds. (I believe they had 2 that weren't solvable somehow though so maybe it was a human solving them and it was a typo or mistake)
Also, I was just checking around to see if there were any good methods for telling if a puzzle is solvable without solving it. Seems geeks for geeks have some code for it.
https://www.geeksforgeeks.org/dsa/check-instance-15-puzzle-s... The only other solution I can think of is detecting both configurations (blank in bottom right or top left) and displaying something when either is reached.