How does this card trick work mathematically?

772 Views Asked by At

Shuffle the deck.

Flip over the first $26$ cards (half the deck), face-up, in a pile (must keep them in order)

As you're doing this remember the $7$th card. Take those first $26$ cards and put them on the bottom of the deck. Now flip over the top three cards (next to each other). Count to $10$ with each one. (If you flip a $3$, $K$, $6$, you place $7$ cards from the deck on the three, $0$ on the king, and $4$ on the six. All face cards count as $10$, and the Ace counts as $1$. Add up the three cards you just flipped face up. (In the example it would equal $19$).

From the deck you are holding, the $7$th card from the initial flipping of $26$ will be the $19$th card (or whatever the sum of the $3$ cards you count to ten with is).

How does this card trick work mathematically?

Link: See this Youtube video

1

There are 1 best solutions below

3
On BEST ANSWER

This works because in the steps

  • flip over the top three cards
  • count to 10 with each one
  • add up the three cards and flip that many cards

you're always flipping exactly 33 cards. This puts you at the 7th card of the second half of the deck.

(If you call the value of top three cards $x_1$, $x_2$, and $x_3$, you're flipping $3$ cards in the first step, $(10-x_1) + (10-x_2) + (10-x_3)$ in the second step, and $x_1 + x_2 + x_3$ cards in the last step. This gives a total of 33 cards, independent of the actual value of $x_1$, $x_2$, and $x_3$.)