By tapping Beasly's Flower, which can be found in said shop and at the entrace to Chinatown, the player can access their store of lost puzzles.
Chapter Seven: Chinatown
At a later point in the game, Layton and Luke, now traveling also with Future Luke and Flora, found a peculiar house at the dead end of Azure Avenue in Chinatown.
Inside, Layton and co. met Beasly again. Layton and Luke talked to Beasly to find out more about him and his backstory, and it was here where Beasly, before he could tell his "gripping tale", was swatted (but not killed) by Puzzlette, who in turn replaced him as puzzle keeper.
He could be seen in the credits, wrapped in bandages and cursing about his ill treatment.