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# Card Trick based on Pigeon hole principle

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### Card Trick based on Pigeon hole principle

1. 1. J.M.S.S.Silva
2. 2. Combinatorial Card Trick A magician asks an unsuspecting observer to randomly choose five cards from a standard deck of playing cards. The participant does not show these cards to the magician, but does show them to the magician’s accomplice. The accomplice looks at the five cards, chooses four of them, and shows these four to the magician in a certain ordered manner. The magician immediately identifies the fifth hidden card.
3. 3. Explanation Note that in any hand of five cards there must be two cards of the same suit (by Pigeonhole Principle). The first card that the accomplice shows to the magician is one of these two cards. The other card of the same suit is never shown – it is the mystery card, the card which the magician must discover. Thus, the accomplice can easily communicate the suit of the hidden card.
4. 4. Explanation From those two cards of the same suit, A and B, the accomplice shows the magician card A such that distance(A, B) is 6 or less. Using Pigeonhole principle it can be shown that there is such an arrangement for any two cards.
5. 5. Example For example, given the choice between the three of clubs and the Jack of clubs, the accomplice reveals the Jack. Distance (Jack ,3) = 5 and Distance(3, Jack)= 8. The three of clubs remains hidden.
6. 6. Explanation Finally, the accomplice arranges the last three cards to encode a number from 1 to 6 – the distance from the value of first card to that of the hidden card. A quick calculation allows the magician to discover the value of the mystery card. This is done by decoding the sequence of shown cards.
7. 7. Explanation There are three more cards left. These cards can be arranged in 6 different ways. Based on a numbering method, the 3 remaining cards can be ranked 1st, 2nd and 3rd. The accomplice may agree with the magician that each arrangement of these 3 cards represent a specific number between 1 and 6.
8. 8. Example Below table is an example of a decoding method. Arrangement Encoded Number 1, 2, 3 1 1, 3, 2 2 2, 1, 3 3 2, 3, 1 4 3, 1, 2 5 3, 2, 1 6
9. 9. Explanation The the magician will decode the above sequence and add the decoded number to the first card. This will be the hidden card.
10. 10. Question If following cards are selected what cards have to be shown to the magician and in what order ?
11. 11. Answer First card should be 8 of Hearts. There are two cards from Hearts. Distance (8,Jack)=3 . But Distance (Jack,8)=10. Therefore 8 is showed first.
12. 12. Answer Now we have to encode number 3 using the remaining 3 cards. Using the previous table the arrangement should be 2,1,3 according to their ranks. Below is the correct order.It is assumed that A is the lowest ranked card.
13. 13. Thank You