Futility Closet posted a how-to for this neat mathematical card trick by New York magician Henry Christ. It’s called All Roads.
Shuffle a deck thoroughly and deal out nine cards in a row, face down. Choose a card, look at it, and assemble the nine cards into a stack face down, with the chosen card at the top. Add this stack to the bottom of the deck.
Now deal cards one at a time from the top of the deck into a pile, face up, counting backward from 10 as you do so. If at some point the card’s rank matches the number said, then begin dealing into a new pile at that point, counting again backward from 10. If you reach 1 without a match occurring, then “close” that pile by dealing a face-down card onto it, and start a new pile.
Keep this up until you’ve created four piles. Now add the values of any face-up cards on top of the piles, count down through the remaining cards until you’ve reached this position, and you’ll find your chosen card.
This works because it always leads to the 44th card in the deck, but it takes some thinking to see this. You can put a sealed deck into a stranger’s hands and direct him to perform the trick himself, with mystifying results.
I am in awe of people who can think of things like this.