 Expansion/Analysis
of a Card Trick Comprised of Transformations in 2-Dimensional
Matrices
Aaron Sherbany,
Clarkstown North High School, NYThis
paper illustrates the properties of a card trick which is
commonly called, "The 11th Variation." The card
trick, which involves transformations in 2-dimensional matrices,
is also expanded and explained with variations in this paper.
Certain predictions as to the parameters that govern the
outcome of the trick are made. Possible application is also
suggested.
The
original card trick ("The 11th Variation") is
merely the foundation of this paper. The classic example
of this card trick employs 21 cards and requires 3 rearrangements
of the cards. The goal of the card trick is to "guess"
which of the cards a person is thinking of. After some repeated
rearrangement of the cards, no matter where in the matrix,
the card always ends up as the 11th card or the middle card
in the matrix.
In
order to find out how the trick worked the standard card
trick was examined. First, 21 cards (as in the standard
card trick) are laid out into 3 columns of 7 cards in the
following order:
|
1
|
8
|
15
|
|
2
|
9
|
16
|
|
3
|
10
|
17
|
|
4
|
11
|
18
|
|
5
|
12
|
19
|
|
6
|
13
|
20
|
|
7
|
14
|
21
|
A person
is asked to choose any one of the 21 cards and tell you
only which column it is in. In this example, the chosen
card is 8. Each of the columns is collected in order
into three separate piles so that the bottom card in
each column is on top and the top card in each column is
on the bottom. The piles are stacked so that the pile
with the "chosen column" (which contains the chosen
card) is placed between the other two. The deck is turned
upside down so that the back of the cards are showing. The
cards in the stack (which were previously arranged in columns)
are rearranged in rows. The cards should now be juxtaposed
in the following fashion:
|
1
|
2
|
3
|
|
4
|
5
|
6
|
|
7
|
8
|
9
|
|
10
|
11
|
12
|
|
13
|
14
|
15
|
|
16
|
17
|
18
|
|
19
|
20
|
21
|
Once
again, the person is asked to find their chosen card and
only tell you the location in terms of which column it is
in. The cards are collected in the same order and fashion
as before. The new "chosen column" (containing
the chosen card) is placed in a pile between the other two
piles and the deck is flipped (the same procedure as before).
Rearranging the cards in rows again will produce the following
layout:
|
1
|
4
|
7
|
|
10
|
13
|
16
|
|
19
|
2
|
5
|
|
8
|
11
|
14
|
|
17
|
20
|
3
|
|
6
|
9
|
12
|
|
15
|
18
|
21
|
Again,
the person is asked which column is the "chosen column."
Column 1 is placed between 2 and 3, the deck is flipped,
and the cards are laid out in rows. Now, the middle card
(11th card) in the matrix is the chosen card.
|
4
|
13
|
2
|
|
11
|
20
|
9
|
|
18
|
1
|
10
|
|
19
|
8
|
17
|
|
6
|
15
|
7
|
|
16
|
5
|
14
|
|
3
|
12
|
21
|
Note
that in the previous steps, the chosen card progressively
moved toward the middle. In this example (7 row by 3 column
configuration), the person conducting the trick must ask
for the "chosen column" at least three times to
be sure that the chosen card is in the middle of the matrix.
I proved this by trying the trick with each of the 21 cards.
Depending on their position, the cards required asking numbers
from 1 to 3 to arrive at the center position. Once achieved,
the chosen card will remain at the middle position no matter
how many times the conductor asks. A minimum of 3 rearrangements
is required to guarantee the chosen card to be the 11th
card (middle card).
Four
important questions come to mind that serve as a basis for
expansion and variation of the trick:
- Is
it possible for this trick to work with more or less than
21 cards?
-
Is it possible to make the card appear in another position
of the matrix?
-
Is it possible to use an even number of rows and columns?
-
Can this trick be expanded into 3 dimensional matrices?
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(Editor's
note: A printable
version of this article is available. You
will need Adobe
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