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Counting for Fun Lecture 18: Nov 21 (based on slides in MIT 6.042)

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Guess Your Card (1)

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Guess Your Card (2) (♠ ♥ ♦ ♣)(♠ ♥ ♦ ♣)

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The Magic Trick Students choose 5 cards Assistant reveals 4 of them Magician announces 5 th card

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Decide the order of the 4 cards: 4! = 24 orderings -- but 48 cards remain Decide which 4 cards to list The Magic Trick

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list must come from hand ? 5-card hands (no order) 4-card lists (ordered) Which one to pick? Match hands with 4-Card lists

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4-card lists (ordered) ? How can we ensure consistency? 5-card hands (no order) Match hands with 4-Card lists

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4!

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The graph satisfies Hall’s theorem so there is a match that magician and assistant can use even works for bigger decks Hall ’ s Theorem

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hands to match to lists How will magician & assistant learn them? A Memorable Matching?

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Among 5 cards chosen: at least 2 have the same suit (Pigeonhole Principle) Assistant lists one of them 1 st Aha! The first card has the same suit as the hidden card! Magic Trick Revealed (I)

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Aha! Look at the order of the other 3 cards! How does magician figure out the value of the hidden card? Magic Trick Revealed (II)

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Fix ordering of the deck A ♣ < 2 ♣ < 3 ♣ < … < K ♣ < A ♦ < 2 ♦ < 3 ♦ < … < K ♦ < A ♥ < 2 ♥ < 3 ♥ < … < K ♥ < A ♠ < 2 ♠ < 3 ♠ < … < K ♠ Magic Trick Revealed (II)

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Possible orders for the remaining 3 cards: { SML, SLM, MSL, MLS, LSM, LMS } Magic Trick Revealed (II)

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Of two cards with the same suit, choosing which to reveal can give 1 more bit of information! Aha! Of two cards with the same suit, choosing which to reveal can give 1 more bit of information! Aha! Wait! Only have 6 lists of the remaining 3 cards, but 12 possible hidden cards of the known suit! Magic Trick Revealed (II)

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Clockwise Distance 58 AQ 67 K 2J The smaller clockwise distance between 2 card values is at most 6: 7 6 Hide card with smaller offset. Reveal the other card

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Magic Trick Revealed (Finally) Offset given by order of remaining 3 cards: SML = 1, SLM = 2, MSL = 3, MLS = 4, LSM = 5, LMS = 6. The first card determines the hidden suit ( ♠ ♥ ♦ ♣ ). Hidden value (A … K) = first-card value + offset (≤ 6).

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Example Hidden:First: Offset = 1 = SML: same suit

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Trick can ’ t work with 4-card hands Assistant can reveal Students can pick possible 4-card hands possible 3-card lists

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