The Neatest Trick of All Times

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The Neatest Trick of All Times

• Terry Schoppenhorst
• Springfield Southeast High School
• NCTM Central Region Conference
• Chicago, IL

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How to Set Up the Trick
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Choose a suit

• Choose a (or the) suit that is repeated.
• This will become the key suit and will ultimately
  be the suit of the hidden card.

• By choosing five cards, you are guaranteed to get
  at least one of them to repeat.
• Pigeonhole Principle.

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Choose which card to hide

• Consider the sequence ace through king.
• When you reach the king, start the sequence over.

• Pick the card that is, at a maximum, 6 steps
  above the card of the same suit.
• That card is hidden, while the other becomes the
  key card.

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Determine the order of the four remaining cards

• A 1
• 2 2
• 3 3
• etc.
• J 11
• Q 12
• K 13

• Add the values of the remaining cards. Call this
  value t.
• Find t mod 4.
• This will be the position of the key card
  assuming we use position 0 to position 3.

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Determine the order of the four remaining cards
(continued)

• Consider the sequence clubs, diamonds, hearts,
  and spades (alphabetical order)
• Ignoring the key card, rank the other cards
  by suit into low, medium, and high.

• If there are duplicates within suits use the
  ranking of ace through king.
• LMH 1
• LHM 2
• MLH 3
• MHL 4
• HLM 5
• HML 6
• Permutations.

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Determine the order of the four remaining cards
(continued)

• Place those three cards around the key card
  in the appropriate order.

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How to Guess the Hidden Card
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Determine the key card

• Add up all visible cards using the numbering
  system. Call this number t.
• Find t mod 4.

• This will give you the position of the key
  card.
• Remember to count beginning with 0.

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Determine how much to add to the key card

• Rank the three remaining cards using low, medium,
  and high.
• Add the amount to the key card.

• Remember to wrap around the king, if necessary.

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State the answer

• Amaze all those looking on.

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Time for Practice
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Variations on the Original Trick
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Suppose the Cohort Behaves Badly
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Determine the hidden card

• Exactly the same as original trick.

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Determine the location of the key card

• Determine the distance from the key card to
  the hidden card.
• If 1 3, put two of the remaining cards face
  down.
• If 4 6, put one of the remaining cards face
  down.

• Using the visible cards, determine the location
  of the key card in the same manner as before.

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Determine the location of the three remaining
cards

• UDD 1
• DUD 2
• DDU 3
• DUU 4
• UDU 5
• UUD 6
• Permutations.

• Depending on the number of steps required to get
  to the hidden card, place the cards in the
  appropriate order around the key card.

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Guess the hidden card

• The same steps as the original using the new
  permutation table.

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Time to Practice Again
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Suppose the Cohort Behaves VERY Badly
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Main differences

• Only three cards will be showing and not
  necessarily all face-up.
• If all of them are face down, the hidden card is
  the king of spades.

• Supersuits are required
• Alpha ace of clubs to 4 of diamonds.
• Beta 5 of diamonds to 8 of hearts.
• Gamma all others.

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Main differences (continued)

• The first card face up represents the supersuit
  of the hidden card.

• UDD 1
• DUD 2
• DDU 3
• DUU 4
• UDU 5
• UUD 6
• UUU 7 or 8
• Permutations

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Main differences (continued)

• If all three are face-up, the second and third
  cards would be ranked.
• LH 7
• HL 8
• Permutations

• Amaze onlookers even more.

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Time to Practice Yet Again
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Suppose the Audience Behaves Badly
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Huge differences

• It is possible to allow the audience which card
  to hide.
• The mathematics is not as nice and requires both
  the cohort and the performer to order every
  single card of the entire deck.

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References

• http//www.spelman.edu/colm/fitch.pdf
• http//people.brandeis.edu/kleber/Papers/card.pdf
  
• http//www.diophante.fr/pages/CardShaiSimonson.pdf
  
• http//www.maa.org/columns/colm/cardcolm200502.htm
  l

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Access to software and presentation

• http//www.springfield.k12.il.us/teachers/tschop

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Questions