This is a great problem-solving investigation that is accessible to students in grades 5-8+. The goal is not a formal proof of how or why this works. That would require some advanced understanding of set theory, I believe. Instead, focus on the reasoning students bring to the table about how the selected card moves throughout the performance of the trick. Some interesting ideas that that students can test will come from this.

The video of a performance of this trick can be found below.

I always introduce this by giving pairs or groups of students a group of 9 cards, then run through the steps as seen in the video, above and written below. Math helps me solve the problem and find everyone’s card at the same time.

1. Make sure you have nine cards with no jokers
2. Deal the cards into three piles of three cards each.
3. Pick up any pile and place it, face-down, in your hand.
4. Pick up any other pile and place, face-down, on top of the cards already in your hand.
5. Pick up the last pile and look at the bottom card, then place it, face-down, on top of the cards in your hand.
6. Deal one card to the table, face-down, into a pile, for each letter in the first name (value) of your card.
7. Place all of the remaining cards in your hand on top of the pile you just dealt to the table.
8. Pick up the cards.
9. Deal one card to the table, face-down, into a pile, for each letter in the middle name (“of”) of your card.
10. Place all of the remaining cards in your hand on top of the pile you just dealt to the table.
11. Pick up the cards.
12. Deal one card to the table, face-down, into a pile, for each letter in the last name (suit – the suits all end with the letter “s”) of your card.
13. Place all of the remaining cards in your hand on top of the pile you just dealt to the table.
14. Pick up the cards.
15. MATH will solve the problem. Deal one card to the table for each letter in “MATH.”
16. Turn over the top card in your hand. It will be your thought-of card.

The lesson:

Pairs or small groups of students should have a small group of cards to use as they investigate this trick. Some students will intuitively “mark” the selected card by flipping it face up so they can watch what happens to it as they run through the trick. When this happens, have them share their idea. Others may ask to mark it with a sticker or even a Post-it.

Students determine rather quickly that no matter what the card is:

• it moves from the third card from the top to the seventh card from the top (third from the face) after spelling the first name.
• after spelling the middle name “of,” the selected card moves from the seventh card from the top (third from the face) to the center card (fifth from the top).
• spelling the last name of the card does not move the card at all. It stays in the center of the stack.

Some questions to keep students thinking after these discoveries:

• What is it about the first names (the values) that make every card move from third from top to third from bottom?
• What is it about the word “of” that makes this trick work?
• What is it about the last names of cards (suits) that keeps the card in the same place?

To close the lesson, ask students to share their findings. What is it about the names that makes this work. What would happen if we didn’t use the middle name? Is there a way we could still make it work?

The main goal for this problem is to help students build stamina in problem-solving. Once students figure out a way to “mark” their chosen card to watch it, the rest comes fairly quickly until you keep the problem going by asking further questions about the names (bulleted list above).

The key findings students will be similar to some of these:

• The card value reverses the order of (at least) the top three cards so that (at least) these three cards go from first, second, and third positions to ninth, eighth, and seventh positions, respectively.
• The word “of” just adds two cards to the bottom of the pile, moving the selected card up in the stack from seventh to fifth (the center card of the stack).
• The last name (suit) – all suits have at least 5 letters, so the order of at least the first 5 cards is reverse, but since there are 9 cards, the fifth card will not change position.