## co.combinatorics – "Best Card Trick" Extensions

Kleber's best card trick is as follows: The mark (viewer) chooses freely five playing cards from a standard stack of $$52$$ and pass these five on to the wizard's assistants. The assistant studies these cards and returns one mystery card until the mark and places the remaining four exposed cardsin an order on a table. The magician enters, inspects the order of the cards revealed, and then announces the complete identity of the Mystery card.

The trick works through the skillful use of mathematics. The five cards selected must contain a color represented by two (or more) cards. The wizard chooses one of them as the Mystery card and another of the same color first in the order of the exposed cards. This is how the magician learns the color of the Mystery Card.

Playing cards can be placed in canonical order ($$clubsuit A, 2, ldots K, diamondsuit A, 2, ldots K,$$ etc.) and thus the three remaining exposed cards can be inserted $$3!$$ possible order of the order. Thus, the assistant can signal six candidate card values, which are counted from the value of the first card (modulo 13). However, this approach alone does not cover all 12 potential map values. This problem is avoided by the wizard being careful Which The card of a pair of the same color is returned and used as the first card in the sequence: use as the first card the card whose value is less than seven steps before the other of the same color (modulo 13) is the secret card; that way the $$3!$$ possible steps ensure that the Mystery card can be reached from the first card in the exposed sequence.

Question 1: How many four-card-illuminated sequences can occur in such tricks?

Question 2: How many groups of five selected cards have more than one acceptable sequence of exposed cards?

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