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The Neatest Trick of All Times Terry Schoppenhorst Springfield Southeast High School NCTM Central Region Conference Oklahoma City, OK

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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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Practice Setting Up the Trick

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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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Practice Setting Up the Trick Again

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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. Make sure the hidden card is a maximum of 8 away from the “key 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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Practice Setting Up the Trick Yet Again

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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 and access to software and the presentation http://www.springfield.k12.il.us/teachers/tschop

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Questions

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