I made a version of Mahjong Solitaire for Yahoo! Games back in the early aughties. I was stuck in figuring out how to make each board solvable. I asked Udi Manber, and he obviously in hind sight, was like, "why don't you just place them in pairs to set up?"
Relatable! My first version of the tile generation placed every tile randomly, which meant I always got stuck, haha. Setting the board up in pairs is indeed what solved the issue.