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## Question

The Card Trick

I ask Alex to pick any 5 cards out of a deck with no Jokers.

He can inspect then shuffle the deck before picking any five cards. He picks out 5 cards then hands them to me (Peter can't see any of this). I look at the cards and I pick 1 card out and give it back to Alex. I then arrange the other four cards in a special way, and give those 4 cards all face down, and in a neat pile, to Peter.

Peter looks at the 4 cards i gave him, and says out loud which card Alex is holding (suit and number). How?

The solution uses pure logic, not sleight of hand. All Peter needs to know is the order of the cards and what is on their face, nothing more.

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The first card laid face down indicates the suit by the position you lay it lengthwise (i.e. East-West, North-South, Northeast-Southwest, Northwest-Southeast). Since the card is rectangular, is shouldn't be too difficult to accomplish. This is shown below:

- Diamonds

| Club

/ Hearts

The next step is the number. The top left corner will be +1, and it will increase clockwise. Overlap the corner of the original card that corresponds to the amount. Using the 3 remaining cards, you can show up to a total of 12 (i.e. Queen). If the card is a King, a the positioning of the final card (East-West or North-South) can indicate whether it is a King or Queen.

I know this sounds confusing, but I think it should all work out. Chances are there is a simpler system though...

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The Card Trick

I ask Alex to pick any 5 cards out of a deck with no Jokers.

He can inspect then shuffle the deck before picking any five cards. He picks out 5 cards then hands them to me (Peter can't see any of this). I look at the cards and I pick 1 card out and give it back to Alex. I then arrange the other four cards in a special way, and give those 4 cards all face down, and in a neat pile, to Peter.

Peter looks at the 4 cards i gave him, and says out loud which card Alex is holding (suit and number). How?

The solution uses pure logic, not sleight of hand. All Peter needs to know is the order of the cards and what is on their face, nothing more.

May I assume that the back of the card is not a symmetrical pattern?

The card chosen is a suit doublet and the first of the 4 cards remaining is the same suit. The 4 cards have values of 1,2,4,and 8. Their right side or upside down orientation determines if their value is true or false. Peter just then sums the truth value products to determine the card chosen.

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The Card Trick

I ask Alex to pick any 5 cards out of a deck with no Jokers.

He can inspect then shuffle the deck before picking any five cards. He picks out 5 cards then hands them to me (Peter can't see any of this). I look at the cards and I pick 1 card out and give it back to Alex. I then arrange the other four cards in a special way, and give those 4 cards all face down, and in a neat pile, to Peter.

Peter looks at the 4 cards i gave him, and says out loud which card Alex is holding (suit and number). How?

The solution uses pure logic, not sleight of hand. All Peter needs to know is the order of the cards and what is on their face, nothing more.

I believe there are many ways to do it, but here is my version.

(For reference, Ace = 1, Jack = 11, Queen = 12, King = 13)

The cards are piled one on top of another, each individually either face-up or face-down. The pile is oriented, and passed to Peter.

Peter can figure out the suit by orientation (length-wise) of the pile:(using an xy coordinate system) x=0 for hearts, y=0 for diamonds, diagonal along y=x for spades, diagonal along y=-x for clubs.

Next, he looks at the 1st card. If the top card is face up, then he knows it is an odd card (A,3,5,7,9,J,K). Face down, even card (2,4,6,8,10,Q). He takes the remaining possibilities, and assigns them a new value upon this system (new value being n):

k={2n-1} or k={2n}

n|k or n|k

1|1 1|2

2|3 2|4

3|5 3|6

4|7 4|8

5|9 5|10

6|11 6|12

7|13

The 2nd card acts the same way the previous card does with the new values (n). It either eliminates the odd values, or the even values.

He repeats the process with the 3rd card, and repeats again for the 4th card, deducing the suit and value.

Spoiler for Example::

The card given to alex is the Ace of Spades (sing it).

The pile is handed to Peter.

Peter inspects the pile and sees that it is diagonal along y=x. (Spades)

Next, he looks at the first card. Face up. So the card Alex is holding is either: 1,3,5,7,9,11,13

Second card: face up. Alex is holding: 1,5,9,13

Third card: face up. Alex is holding: 1,9

Fourth card: face up. Alex is holding an ace.

Try another combination. In some cases, he could know by the third card (such as an even case on either the first or second card).

This should work for any standard-shaped (rectangle) deck of cards.

Edited by tomaketu

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