some variations on a mathematical card trick
TRANSCRIPT
Some Variations on a Mathematical Card TrickAuthor(s): LEE M. MARKOWITZSource: The Mathematics Teacher, Vol. 76, No. 8 (November 1983), pp. 618-619, 577Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27963722 .
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Some Variations on a
Mathematical Card Trick By LEE M. MARKOWITZ, Bowling Green State University, Bowling Green, OH 43403
In
" A Simple Mathematical Model of a Neat Card Trick," John Elliott (1982)
explained a trick that has appeared in sev
eral books (Patton 1968* ; Scarne 1950). It is an intriguing activity that can be gener alized in several ways. Let's first review the
procedures, and then see how we can gener alize from them.
Recall that piles of cards are formed by dealing a card faceup and then using that card to initiate a count. Additional cards are dealt faceup on the initial card in such a way that each card dealt advances the count by one. When the count reaches thir
teen, dealing is discontinued, and the pile of face-up cards is flipped over so that the card that initiated the count is now the top card of a face-down pile. For example, if a 9 was dealt faceup initiating the count, four additional cards would be dealt faceup ad
vancing the count to 13. The pile of five
face-up cards would then be flipped face down so that the 9 would be the top card of a face-down pile. Piles are formed in this manner until the fifty-two cards in the deck have been dealt or until not enough cards remain to form an additional pile. The sum of the top cards of any three piles equals ten plus the number of cards not in the three piles. Thus, if the top cards of two of three piles are known, along with the number of cards not in the selected piles, then the top card of the third pile can be
determined. Elliott invited readers of his article to
explore variations of this trick. It can be
generalized in three ways: (a) the number
*Temple Patton's book is devoted exclusively to math ematical card tricks. A mathematical proof is included with each trick. In addition to the mathematical principle that is the basis of the first trick described, Patton discusses how the commutative law, base-two arithmetic, and a special property of the number 9 (that the sum of the digits of a multiple of 9 is a multiple of 9) can be employed in fascinating card tricks.
ending the count for each pile dealt need not be thirteen ; (b) the number of piles se
lected need not be three; and (c) the number of piles whose top card is revealed need not be one less than the number of
piles selected. If two or more top cards are
not revealed, their sum can be determined. The following formula incorporates all
three generalizations :
(I), +
+ ?(A + l- al) = 52>
where
k is the number ending the count,
is the number of piles selected,
a,, dj-i are the values of the unre
vealed top cards of piles,
dj, ..., ap are the values of the revealed
top cards of piles, and
is the number of cards that are not in the piles selected.
Note that the two summation terms in
(1) represent the number of cards in the se
lected piles whose top cards are either unre
vealed or revealed, respectively. We now
solve for the sum of the unrevealed top cards,
/i-i \
Some algebraic manipulation of (1)
yields
52- kp-p + |> (2) =
All the information necessary to determine the sum of the unrevealed top cards is
available. For example, assume piles are
formed with the count ending at fourteen for each. Next, assume four piles are select
618 -Mathematics Teacher
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ed by a spectator. At this point, the per former mentally calculates the value of
52 - kp - = 52 - (14)(4)
- 4 = '8.
Next, assume that two top cards, a 9 and a
10, are turned over by the spectator. The
performer than adds their sum, 19, to "8,
yielding 11. Next, the performer counts the
number of cards that are not in the four
selected piles. Assume this number equals 24. Then the performer subtracts 11 from 24
and tells the spectator that the sum of the
two unrevealed top cards is 13.
A New Trick
I would like to present another neat math
ematical card trick whose algorithm also can be proved by employing algebraic prin
ciples. Assume a stack of an odd number of
cards has been manipulated so that a spec tator's card is positioned in the middle of
the stack. One way to do this is to have a spectator
select a card, set it aside, and then form
four piles with an equal number of cards
from a subset of the remaining fifty-one cards. Cards not used in the four piles are
not used in the trick. The spectator places his selected card on the top or bottom of
any pile. The performer first picks up a pile that the spectator has not chosen. If the
spectator's card is on the bottom of a pile, it is picked up second; if it's on the top of a
pile, that pile is picked up third. The spec tator's card will then be in the middle of a
stack of cards.
Next, have the spectator neatly deal the stack of cards facedown from left to right into three piles until the stack is used up. The spectator is to note the pile on which the last card falls. If it is the first pile, then that pile is used for the next step ; if it is the second pile, then the third pile is kept (used for the next step) ; and if it is the third pile, then the second pile is kept. The identical
procedure is repeated with the stack that is
kept. The procedure is continued until one
card remains. That card is the spectator's card.
The proof of this algorithm is left to the reader. Here are a few hints to help get you
started. First, show that the algorithm selects the pile with the middle card. Let
t be the total number of cards, m be the number of cards above or below
the middle card including the middle
card,
/ be the number of the pile with the last
card, and
k be the number of cards in the pile with the last card.
Then,
(3) m = (t + l)/2.
(4) f = 3(?fe-l) + ?; ? = 1,2,3.
Now, obtain an expression for m by substi
tuting (4) into (3). Then use the fact that t must be odd and that m must be an integer to show that for integers s2, and s3,
(a) if i = 1 then m = 3sx + 1,
(b) if i = 2 then m = 3s2 + 3, and
(c) if i = 3 then m = 3s3 + 2.
Next, use the variables that have been de fined to show that an equal number of cards is above and below the middle card in the
pile with the middle card.
Having proved this algorithm, try to
prove the following more general one. After a spectator's card has been positioned in the middle of a stack of cards, deal the stack into piles, where is any odd number. Note the number of the pile on which the last card falls. If it is odd, add 1 and then divide by 2. If it is even, add
(x + 1) and then divide by 2. Call the value obtained and keep the pth pile. Repeat this procedure until one card remains. That card is the spectator's card. Note that if
= 3, this algorithm is the same as the one
originally described. For those interested, a
proof of the generalized algorithm appears in Markowitz (1981).
REFERENCES
Elliott, John C. "A Simple Mathematical Model of a Neat Card Trick." Mathematics Teacher 75 (April 1982):308-9.
Markowitz. Lee. "Some Math Magic." Journal of Rec reational Mathematics 13 (Winter 1981): 14-17.
(Continued on page 577)
November 1983-?:-?-619
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his office each day between classes so that the male students could not complain that the hallways were cluttered with females !
Winston stayed at G?ttingen for three
years and finished her dissertation, "Rie mann's Case of Lane's Differential Equa tion." She passed her examinations with honors in 1896. Her degree was actually
granted in 1897 after her dissertation was
published. She was then made head of the math
ematics department at Kansas State Agri cultural College in Manhattan, Kansas.
"Actually, she was all the department," said her daughter, Caroline Newson Be
shers, in a recent interview. "She may have
had only a student assistant to help her." Winston stayed there three years, and in 1900 she gave up the position to marry
Henry Byron Newson, who was acting head
of the mathematics department at the Uni
versity of Kansas. When her husband died in 1910 at the age of forty-nine of a heart
attack, Mary Frances had no job, no pen
sion, no life insurance benefits, and three small children.
Finding a job was not easy. Finally, in
1913, she accepted an offer from Washburn
College in Topeka, Kansas, twenty-five miles away. She left this position in 1921
after one of the first cases of academic free dom ever investigated by the American As
sociation of University Professors occurred on the Washburn campus. Dr. Newson was one of eight faculty members, and the only woman, to sign a petition in support of the
political science professor who had been dismissed for talking too freely about his
political views with his students. Her new position was at Eureka College
in Illinois, a smaller school than Washburn. "But she was the head of the department, and as had been true at Kansas State, prac
tically the whole show," her daughter testi
fied. She stayed at Eureka until she retired in 1942.
When she was honored at the Women's
Centennial Congress in New York City in
1940, she was not present. It fell on the first
day of classes for a new term at Eureka, and she felt she had to teach her classes. So
she replied to the invitation, " Thank you,
but I can't make it," and she didn't even
mention it to her friends or colleagues. However, the whole community was made aware of the celebrity among them when the local newspaper announced with a bold
headline, gets national honor and tells nobody. Beshers recounted, "Mama said, *Well, you know, I didn't have very many intimate friends there, and there wasn't
anybody I really thought cared.' And that was absolutely typical of Mama. Anybody else would have mentioned it, I'm sure. We realize now how lonely she was most of the time."
Newson spent the last several years of her life with her daughter, Beshers, and died 5 December 1959 at the age of ninety. When her obituary was printed in the
Washington Post, it triggered a response, and papers all over the country carried the article. "It was even on the radio," said her
daughter. "My friends were astounded!
Many of them did not even know that she had gone to college."
BIBLIOGRAPHY
Beshers, Caroline Winston. Interview with B. S. Whit
man, December 14,1981.
Kenschaft, P. C, and Kaila Katz. "Sylvester and Scott." Mathematics Teacher 75 (September 1982):490-94.
Osen, Lynn M. Women in Mathematics. Cambridge, Mass.: M.I.T. Press, 1974.
Perl, Teri. Math Equals. Menlo Park, Calif. : Addison
Wesley Publishing Co., 1978.
"Women & Science" class, University of Washington Women Studies Program. Hypatia's Sisters. Seattle, Wash. : Feminists Northwest, 1976. W
Some Variations on a Mathematical Card Trick (Continued from page 619)
Patton, Temple. Card Tricks Anyone Can Do. New
York: Castle Books, 1968.
Scarne, John. Scarne on Card Tricks. New York:
Signet Books, 1950.
BIBLIOGRAPHY
Davis, Edward J., and Ed Middlebrooks. "Algebra and
a Super Card Trick." Mathematics Teacher 76 (May
1983):326-28. Escultura, Eddie. "A Number Trick?Explained with
Algebra." Mathematics Teacher 76 (January
1983):20-21.
November 1983-?-577
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