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ArsDigita University
Month 2 – Discrete Mathematics and Probability
Professor Shai Simonson
The Combinatorial Card Trick – Challenge Problem
We showed a method in class based on combinatorics, which allows a magician and
an accomplice to perform the following trick.
The magician asks someone to choose five
cards randomly from a standard deck.
The accomplice then looks at these cards, and shows
four of them in a particular order to the magician, who then immediately identifies the last
card.
At first thought the trick seems impossible, because there are only 4! ways to order
four cards, and 24 pieces of information is not enough to identify a unique value among the
possible 48 cards remaining.
However, a more careful analysis reveals that the accomplice
has more choices than 24 up his sleeve.
The accomplice may choose which four cards to
show, as well as their order.
There are C(5,4) ways of choosing four cards from five, hence
the accomplice has 120 different pieces of information he can send.
The hard part is that
the magician cannot easily decode which of these 120 was sent.
She can easily decode 24
pieces of information, but not so easily the extra factor of five.
One method that allows the magician to decode more than the obvious 24 is to
notice that among the five cards chose, there must be two of the same suit (due to the
pigeonhole principle).
The first card shown by the accomplice is one of these two cards,
and the second is never shown.
If we number the cards in a suit from 1 to 13, and do
arithmetic modulo 13, then given any two cards, there is always one which is six or less
below the other (again the pigeonhole principle – if they were both 7 or more less than the
other, then there needs to be 14 cards).
The accomplice chooses the card that is 6 or less
below the other (modulo 13).
For example, given the 3 and the Jack, we choose the Jack.
The accomplice then chooses the order of the next three cards to encode a number
from 1 to 6.
The magician decodes this number the last three cards, (value 16) and adds it
to the first card, in order to recover the identity of the missing card.
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 Spring '09

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