On the 21 Card Trick and Verini's Lost Trick

We provide a necessary and sufficient condition for the solvability of generalized variants of the twenty-one card trick (21CT). The 21CT works by dealing the cards into three stacks several times and, after the spectator identifies the stack containing their card each round, arranging the stacks so the performer always knows where the chosen card ends up for the final reveal. Our analysis extends to any number of cards, arranged in multiple stacks, with an arbitrary number of cards per stack and an arbitrary number of iterations of dealing into stacks. We also allow for flexible collection of the stacks induced by face-down or face-up dealing of the deck of cards, ensuring that the unknown card appears at a predetermined position within the deck. Notably, this also allows us to analyze the historically first card trick to appear in print, proposed by Verini in 1542---a lost card trick that was rediscovered only recently: it turns out that the trick does not work. But, we also demonstrate how it can be fixed.