...

ALGEBRA AND A SUPER CARD TRICK Source:

by user

on
Category: Documents
12

views

Report

Comments

Transcript

ALGEBRA AND A SUPER CARD TRICK Source:
ALGEBRA AND A SUPER CARD TRICK
Author(s): EDWARD J. DAVIS and ED MIDDLEBROOKS
Source: The Mathematics Teacher, Vol. 76, No. 5 (May 1983), pp. 326-328
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/27963518 .
Accessed: 14/10/2014 13:30
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected]
.
National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend
access to The Mathematics Teacher.
http://www.jstor.org
This content downloaded from 155.33.16.124 on Tue, 14 Oct 2014 13:30:33 PM
All use subject to JSTOR Terms and Conditions
dues
ALGEBRA
AND A SUPER
By EDWARD J.DAVIS
University
of Georgia
Athens, GA 30602
and ED MIDDLEBROOKS
CARD TRICK
ber one card. Then continue the dealing
process so that you place exactly tenmore
cards faceup on top of your secretly select
ed card (see fig. 1).
First Presbyterian
Day School
Macon, GA 31204
is a fascinating card trick that can
be explained and justified using first-year
algebra. We have used this trickwith high
school classes and mathematics clubs as a
device and sometimes as a
motivational
to
students to find or finish the
challenge
procedure
algebraic
justification. This
makes a good example of the power of
mathematics to unmask seemingly complex
situations and, therefore, is a good device
for teachers to "keep up their sleeves" for
some auspicious occasion or a time when
interest is lagging. Some related procedures
can be found in the bibliography.
Get a standard fifty-two-card deck and
work through each step. Later we shall
examine the algebra behind the scenes. We
assume that jacks, queens, and kings have
Here
values of 11, 12> and
13, resp?ctively,
whereas aces have a value of 1.
1. Shuffle the deck and begifi placing
cards faceup all in one stack on a desk top.
the se
that you are memorizing
Claim
the
quence of cards displayed. Challenge
feat
to
mental
spectators
perform this great
as well !
2. As you are placing cards faceup in
step one (say after you have dealt about a
dozen cards), secretly pick out and remem
326
Mathematics
secretly
'Selected
card
Stack ofc?ids faceup
Fig.
1
3. Have each of three students select one
card at random from the cards in your
hand. Have these three cards placed faceup
in three separate locations on the desk top.
Let's assume the students selected a 4, an 8,
and a jack.
4. Turn over
the face-up pile of cards
containing your secretly selected (and me
morized) card and place it under the stack
of cards in your hand. You now have three
cards, each facing up, and one stack of
cards in your hand.
work with each face-up card
additional cards facedown
Place
separately.
on top of each of the three face-up cards.
Start with the value of each face-up card
and add cards until you reach a count of
thirteen. Let the bottom face-up card stick
out a little so you can use it in the next step
5. Now
(see fig. 2).
Teacher
This content downloaded from 155.33.16.124 on Tue, 14 Oct 2014 13:30:33 PM
All use subject to JSTOR Terms and Conditions
Face-down
cards counted
as shown
Fig.
out loud, add the values
6. Now,
the three face-up cards on the bottom
the piles in view. We shall call this sum
we
5 = 4
have
example
(In our
8+
of
of
S.
there are
4
cards on the table. In step 7 you counted
into the
S = nl + n2 + n$ cards down
cards in your hand. Adding the number of
cards on the table and S gives
11 =23.)
7. Ask ifanyone knows the value of the
Sth card in your hand. Pretend you are
you
struggling to recall it?remember,
a long se
claimed to have memorized
announce it as ifyou
quence of cards?and
were able to remember it! Count out S
(14
(14-nj
+(14-nj+
+ (14-HJ +(14-aj
+
You will always
==
(Wl+ n2 + n3? 42.
reach card number forty
many cards are beneath your se
put them
cretly selected card? Ten?you
How
A Rationale
Let's begin by finding an expression for
the total number of cards in each of the
three piles of cards left in view. We had
piles on top of 8,4, and a jack.
the 8 pile, we had 6 =
(14-nj
two.
cards, and you can't miss!
On
2
14 ?
therein the second step! (See fig. 3.) That
remaining cards
fromyour
hand
8 cards.
On the 4 pile, we had a total of 10 =
14 ? 4 cards.
On
3=
the jack pile, we
14 ? 11 cards.
had
a
total of
In general, if is the value of the face-up
?
cards
card on the bottom, there are 14
in the pile. If nl9 n2, and n3 are the values
of the face-up cards on the bottom, then
yoursecretly
selected card
Stack ofcards facedown
Fig.
3
May 1983
This content downloaded from 155.33.16.124 on Tue, 14 Oct 2014 13:30:33 PM
All use subject to JSTOR Terms and Conditions
327
means
your secretly selected and memo
rized card is also card number forty-two.
Teacher's
Corner
As with all card tricks, a dash of the
atrics can amplify the positive effects. Pre
tending to struggle to recall thememorized
a
card and claiming to have memorized
are
ma
two
cards
of
such
sequence
long
neuvers. Holding
or
off an explanation
limiting the number of "performances" on
a given day are other devices that could
BIBLIOGRAPHY
Felps, Barry C. "An Old Card Trick Revisited."
ematics Teacher 69 (December
1976):665-66.
Math
and Mystery.
Gardner, Martin. Mathematics:
Magic
New York: Dover,
1956. (See chapters 1 and 2.)
Hadar, Nita. "Odd and Even Numbers." Mathematics
Teacher 75 (May 1982):408-12.
J. Davis.
"The 22nd Card
Hays, Katie, and Edward
Trick." Illinois Mathematics
Teacher 30 (September
1979):16-17.
Puzzles
Heath, Royal Vale. Mathemagic:
Magic,
Games with Numbers. New York : Dover,
1953.
Stern, Burton
L. "Algebra
in Card
and
Tricks." Mathemat
icsTeacher66 (October 1973):547.
W.
"A Card
Trigg, Charles
Teacher 63 (May 1970):395-96.
Trick."
Mathematics
interest. If the class is
spark additional
challenged to find a rationale, theywill find
it easier if they know all the steps involved.
Once students know how to do the trick,
they are usually very interested in finding
out why itworks.
Here are some questions and suggestions
we have used to help students find the alge
braic explanation.
Look at the total number of cards in
each of the three piles. How can you pre
dict each total if you know the value of
the bottom face-up card?
TODAY'S TWISTER
A DallyMathEnrichment
Program
180problemsinquadrants
on81/2 1 pages
Proveneffective;
enlivens
gr.6-9math
format
for
of individual
Convenient
Easilyadministered.
supply
creating
twisters.
comments
included
answers,
Program
suggestions,
$8.00 includes
postage
PaulG. Dickie
26 Lynn*Road,Sudbury,
MA01776
Think of the cards when they are in the
three piles as all being in one stack?this
includes the cards in your hand. How far
down in the deck is your secret card?
Here are some questions we have posed
to get students to look back at their ration
ales.
Will the trick always work, or can some
one pick three cards at random that will
cause it to fail?
Suppose we count to fourteen instead of
thirteen as we place cards on top of the
three selected cards. What other change
do we have tomake?
Why is the number of cards in each pile
14 ? n? If we count to 13, it seems as if
we should have 13 ? n.
What could happen if four students se
lected cards in the third step?
What
in the
changes could you make
trick if you had a double deck (104
cards) to work with?
328
Mathematics
attach
_...
\label
UCDC
ncnc
Mail
,fy?uare?neofthemanvNC mem"
bers who will move this year, please let
us know fiveweeks beforechangingyour/
address. Use this form-attach your
magazine address label and printyour
new address below.
to:NCTM
1906 Association Drive
Reston, Virginia 22091
Name_
New
Address_
City_
State.
Teacher
This content downloaded from 155.33.16.124 on Tue, 14 Oct 2014 13:30:33 PM
All use subject to JSTOR Terms and Conditions
.ZIP Code_
Fly UP