Last week, I saw a magic show, and was the volunteer for climactic act. The magician introduced this act by talking about the Fitch-Cheney card trick, but then explained that she was going to do an even more impressive trick. Namely, instead of using a $52$ card deck, she would use a larger deck, with $N$ cards, though she didn't say exactly how big $N$ was.
Here's what happened:
The magician left the room. An assistant asked me to look through the $N$ card deck, and pick out any $5$ cards. The assistant inspected the cards I chose, then gave one back to me, which I put in my pocket. He arranged the remaining four cards in a neat stack, and placed this stack face down on a table. The magician returned, inspected the stack, and successfully guessed the card in my pocket.
What is the largest value of $N$ for which the trick can be performed? For that $N$, how is it done?
Note that there is no slight of hand or secret communication. Each card is distinct, the faces are rotationally symmetric, and the stack is placed at the exact same spot on the table every time the trick is performed. The assistant is only allowed to choose which card the volunteer gets ($5$ choices), then permute the four other cards ($4!=24$ choices).
Aside: From this question, we know $N=53$ is possible.