Shuffle the deck, and then have a Volunteer from the Audience think of a number between 1 and 20. Give the deck to The Volunteer, and tell them to remove that many cards from the deck while your back is turned. If it makes them feel better, they can shuffle the cards again afterwards. The Volunteer should put the removed cards out of sight and not tell you how many cards were taken, although it is fine for The Volunteer to tell the rest of the audience.
The Volunteer now returns the rest of the deck to you. Explain that you are going to deal 20 cards face up as you count from 1 to 20, and the Volunteer should memorize the one that corresponds to the number of cards that they removed. At the end of the trick you will reveal this memorized card to the Audience. (Alternately, instead of returning the deck to you, this step can be done by the Volunteer while your back is still turned.)
Afterwards, ask The Volunteer to pick a number that’s less than however many cards are left and to tell you that number. For this example, let’s say that The Volunteer picks 14. You deal that many cards face up on top of the face-up pile of 20. (Alternately, this step can also be done by the Volunteer, but you do need to know how many cards are put on top of the pile of 20.)
At this point you should take the rest of the cards, and be very careful not to count them, as your skeptical Volunteer will be watching for this. Turn the face-up pile face-down — in our example that pile has 34 cards in it — and put the rest of the cards on top, still face down.
Explain that you will use your incredible mind-reading powers to reveal the memorized card. Quietly to yourself count down through the deck from the top, counting silently 35, 36, 37, ….. (starting one higher than however many cards were in that face-up pile).
The memorized card will be the one that you get to when you reach the count of 52. You can reveal the card without ever seeing it yourself (just hold it up towards the Volunteer) or you can be fancy and take each card from the top, look at it, and decide if it’s right. What’s particularly impressive is that you can figure out the card without ever knowing directly how many cards were removed from the pile!
I found this trick under “Twentieth Century Mind-Reading” here on e-math. The person who wrote it up there attributes it to the book Card Tricks Anyone Can Do (Castle Books, 1968).