The unshuffle and other card tricks

James Carraher

University of Nebraska – Lincoln

s-jcarrah1@math.unl.edu

March 2013

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Trick 1

Today we are going to learn several “magic” tricks.

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Trick 1

Today we are going to learn several “magic” tricks.

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Trick 1

Today we are going to learn several “magic” tricks.

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Trick 1

Today we are going to learn several “magic” tricks.

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Trick 1

Today we are going to learn several “magic” tricks.

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Trick 1

Today we are going to learn several “magic” tricks.

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The next tricks use a special shuffling technique.

The perfect shuffle (or faro shuffle) is performed by splittingthe deck in half and interweaving the two halves.

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The next tricks use a special shuffling technique.

The perfect shuffle (or faro shuffle) is performed by splittingthe deck in half and interweaving the two halves.

3 / 17

The next tricks use a special shuffling technique.

The perfect shuffle (or faro shuffle) is performed by splittingthe deck in half and interweaving the two halves.

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The perfect shuffle is difficult to perform so we are going touse a variation of the perfect shuffle, the unshuffle.

The unshuffle is performed by splitting the deck into two pilesone card at a time and then placing on the of the piles on topof the other one.

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The perfect shuffle is difficult to perform so we are going touse a variation of the perfect shuffle, the unshuffle.

The unshuffle is performed by splitting the deck into two pilesone card at a time and then placing on the of the piles on topof the other one.

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The perfect shuffle is difficult to perform so we are going touse a variation of the perfect shuffle, the unshuffle.

The unshuffle is performed by splitting the deck into two pilesone card at a time and then placing on the of the piles on topof the other one.

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Trick 2

Given the cards ace through queen in order shuffle the deckusing the unshuffle.

How does knowing the bottom card help me find the top card?

The unshuffle preserves central symmetry. The sum of theinside and outside card is always 13.

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Trick 2

Given the cards ace through queen in order shuffle the deckusing the unshuffle.

How does knowing the bottom card help me find the top card?

The unshuffle preserves central symmetry. The sum of theinside and outside card is always 13.

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Trick 2

Given the cards ace through queen in order shuffle the deckusing the unshuffle.

How does knowing the bottom card help me find the top card?

The unshuffle preserves central symmetry. The sum of theinside and outside card is always 13.

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Trick 2

Given the cards ace through queen in order shuffle the deckusing the unshuffle.

How does knowing the bottom card help me find the top card?

The unshuffle preserves central symmetry. The sum of theinside and outside card is always 13.

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Trick 2

Given the cards ace through queen in order shuffle the deckusing the unshuffle.

How does knowing the bottom card help me find the top card?

The unshuffle preserves central symmetry. The sum of theinside and outside card is always 13.

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Trick 2

Given the cards ace through queen in order shuffle the deckusing the unshuffle.

How does knowing the bottom card help me find the top card?

The unshuffle preserves central symmetry. The sum of theinside and outside card is always 13.

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Trick 2

Given the cards ace through queen in order shuffle the deckusing the unshuffle.

How does knowing the bottom card help me find the top card?

The unshuffle preserves central symmetry. The sum of theinside and outside card is always 13.

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Trick 2

Given the cards ace through queen in order shuffle the deckusing the unshuffle.

How does knowing the bottom card help me find the top card?

The unshuffle preserves central symmetry. The sum of theinside and outside card is always 13.

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How well does the unshuffle shuffle the deck?

Total number of ways to shuffle the 12 cards

12× 11× 10× 9× 8× 7× 6× 5× 4× 3× 2× 1 = 12!

= 479001600

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How well does the unshuffle shuffle the deck?

Total number of ways to shuffle the 12 cards

12× 11× 10× 9× 8× 7× 6× 5× 4× 3× 2× 1 = 12!

= 479001600

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How well does the unshuffle shuffle the deck?

Total number of ways to shuffle the 12 cards

12× 11× 10× 9× 8× 7× 6× 5× 4× 3× 2× 1 = 12!

= 479001600

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How well does the unshuffle shuffle the deck?

Total number of ways to shuffle the 12 cards using theunshuffle.

12× 10× 8× 6× 4× 2 = 46080

12× 10× 8× 6× 4× 212!

∼= 0.00009620009.

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How well does the unshuffle shuffle the deck?

Total number of ways to shuffle the 12 cards using theunshuffle.

12× 10× 8× 6× 4× 2 = 46080

12× 10× 8× 6× 4× 212!

∼= 0.00009620009.

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How well does the unshuffle shuffle the deck?

Total number of ways to shuffle the 12 cards using theunshuffle.

12× 10× 8× 6× 4× 2 = 46080

12× 10× 8× 6× 4× 212!

∼= 0.00009620009.

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How well does the unshuffle shuffle the deck?

Total number of ways to shuffle the 12 cards using theunshuffle.

12× 10× 8× 6× 4× 2 = 46080

12× 10× 8× 6× 4× 212!

∼= 0.00009620009.

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How well does the unshuffle shuffle the deck?

Total number of ways to shuffle the 12 cards using theunshuffle.

12× 10× 8× 6× 4× 2 = 46080

12× 10× 8× 6× 4× 212!

∼= 0.00009620009.

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Trick 3

In groups of five deal out 5 cards to each person.

1 Pick a card from the five dealt cards and place it on top.2 Stack all 5 hands back into one pile.3 Go around either cutting the deck or performing the

unshuffle.4 After everyone has either cut the deck or performed the

unshuffle deal the cards back out one at a time.

One person should be dealt all five selected cards.

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Trick 3

In groups of five deal out 5 cards to each person.

1 Pick a card from the five dealt cards and place it on top.2 Stack all 5 hands back into one pile.3 Go around either cutting the deck or performing the

unshuffle.4 After everyone has either cut the deck or performed the

unshuffle deal the cards back out one at a time.

One person should be dealt all five selected cards.

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Trick 3

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Trick 3

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Trick 3

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The perfect shuffle.

The following video is an shows how to perform the perfectshuffle. Performing 8 perfect shuffles results in the deckreturning to its original order. Perfect Shuffle

(http://www.youtube.com/watch?v=OFjgPGPQ0NA)

Label the cards from 0 to 51. Then the position of card aftera perfect shuffle is

2 mod 51 if 0 ≤ ≤ 5051 if = 51 .

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The perfect shuffle.

The following video is an shows how to perform the perfectshuffle. Performing 8 perfect shuffles results in the deckreturning to its original order. Perfect Shuffle

(http://www.youtube.com/watch?v=OFjgPGPQ0NA)

Label the cards from 0 to 51. Then the position of card aftera perfect shuffle is

2 mod 51 if 0 ≤ ≤ 5051 if = 51 .

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Modular Arithmetic

What is 2 mod 51?

Divide 2 by 51 and record the remainder.

For example suppose = 27.

Then 2 = 54 = 51× 1+ 3,

so 54 mod 51 ≡ 3 mod 51.

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Modular Arithmetic

What is 2 mod 51?

Divide 2 by 51 and record the remainder.

For example suppose = 27.

Then 2 = 54 = 51× 1+ 3,

so 54 mod 51 ≡ 3 mod 51.

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Modular Arithmetic

What is 2 mod 51?

Divide 2 by 51 and record the remainder.

For example suppose = 27.

Then 2 = 54 = 51× 1+ 3,

so 54 mod 51 ≡ 3 mod 51.

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The perfect shuffle

After one perfect shuffle, card is at position 2 mod 51.

After two perfect shuffles 2(2) mod 51 ≡ 22 mod 51.

After three perfect shuffles 2(22) mod 51 ≡ 23 mod 51.

After 8 perfect shuffles, card is at position

28 mod 51 ≡ 256 mod 51.

Since 256 = 51× 5+ 1, 256 mod 51 ≡ 1 mod 51.

So 28 mod 51 ≡ mod 51.

So all the cards are in the same position that they started in.

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The perfect shuffle

After one perfect shuffle, card is at position 2 mod 51.

After two perfect shuffles 2(2) mod 51 ≡ 22 mod 51.

After three perfect shuffles 2(22) mod 51 ≡ 23 mod 51.

After 8 perfect shuffles, card is at position

28 mod 51 ≡ 256 mod 51.

Since 256 = 51× 5+ 1, 256 mod 51 ≡ 1 mod 51.

So 28 mod 51 ≡ mod 51.

So all the cards are in the same position that they started in.

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The perfect shuffle

After one perfect shuffle, card is at position 2 mod 51.

After two perfect shuffles 2(2) mod 51 ≡ 22 mod 51.

After three perfect shuffles 2(22) mod 51 ≡ 23 mod 51.

After 8 perfect shuffles, card is at position

28 mod 51 ≡ 256 mod 51.

Since 256 = 51× 5+ 1, 256 mod 51 ≡ 1 mod 51.

So 28 mod 51 ≡ mod 51.

So all the cards are in the same position that they started in.

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The perfect shuffle

After one perfect shuffle, card is at position 2 mod 51.

After two perfect shuffles 2(2) mod 51 ≡ 22 mod 51.

After three perfect shuffles 2(22) mod 51 ≡ 23 mod 51.

After 8 perfect shuffles, card is at position

28 mod 51 ≡ 256 mod 51.

Since 256 = 51× 5+ 1, 256 mod 51 ≡ 1 mod 51.

So 28 mod 51 ≡ mod 51.

So all the cards are in the same position that they started in.

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The perfect shuffle

After one perfect shuffle, card is at position 2 mod 51.

After two perfect shuffles 2(2) mod 51 ≡ 22 mod 51.

After three perfect shuffles 2(22) mod 51 ≡ 23 mod 51.

After 8 perfect shuffles, card is at position

28 mod 51 ≡ 256 mod 51.

Since 256 = 51× 5+ 1, 256 mod 51 ≡ 1 mod 51.

So 28 mod 51 ≡ mod 51.

So all the cards are in the same position that they started in.

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The perfect shuffle

After one perfect shuffle, card is at position 2 mod 51.

After two perfect shuffles 2(2) mod 51 ≡ 22 mod 51.

After three perfect shuffles 2(22) mod 51 ≡ 23 mod 51.

After 8 perfect shuffles, card is at position

28 mod 51 ≡ 256 mod 51.

Since 256 = 51× 5+ 1, 256 mod 51 ≡ 1 mod 51.

So 28 mod 51 ≡ mod 51.

So all the cards are in the same position that they started in.

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Trick 4

Take one card from every suit.

Moves1 You can flip all the cards over.2 Take the top card and move it to the bottom.3 Flip the top two cards over.

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Trick 4

Take one card from every suit.

Moves1 You can flip all the cards over.2 Take the top card and move it to the bottom.3 Flip the top two cards over.

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Trick 4

Take one card from every suit.

Moves1 You can flip all the cards over.2 Take the top card and move it to the bottom.3 Flip the top two cards over.

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Trick 4

Take one card from every suit.

Moves1 You can flip all the cards over.2 Take the top card and move it to the bottom.3 Flip the top two cards over.

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Trick 4

Take one card from every suit.

Moves1 You can flip all the cards over.2 Take the top card and move it to the bottom.3 Flip the top two cards over.

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Trick 4

Take one card from every suit.

Moves1 You can flip all the cards over.2 Take the top card and move it to the bottom.3 Flip the top two cards over.

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Trick 4

Take one card from every suit.

Moves1 You can flip all the cards over.2 Take the top card and move it to the bottom.3 Flip the top two cards over.

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Trick 4

Take one card from every suit.

Moves1 You can flip all the cards over.2 Take the top card and move it to the bottom.3 Flip the top two cards over.

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Trick 5

Take a deck of 20 cards, 10 J Q K A of each suit.

1 I go through the deck in pairs placing some cards face toface, some back to back and others in the same direction.

2 You determine if a pair should be flipped.3 You determine which groups of four cards should be

flipped.4 I deal out the cards.5 You decide how they should be folded back together.

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Trick 5

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Trick 5

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Trick 5

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Trick 5

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Trick 5

Rules The initial flipping of the pairs of cards is all that isneeded to do this trick.

1 If neither card is a heart then place the cards face to face.2 If both cards are hearts then place the cards back to back.3 If there is one heart then both cards are placed faceup,

but the heart always goes on top.

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Acknowledgements:

Unshuffling for the Imperfect Magician. Doug Ensley. MathHorizons.

An Amazing Card Trick. Arthur Benjamin. Harvey MuddCollege.

Chris Aguilar for the library of vectorized playing cards.

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