some variations on a mathematical card trick

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Some Variations on a Mathematical Card Trick Author(s): LEE M. MARKOWITZ Source: The Mathematics Teacher, Vol. 76, No. 8 (November 1983), pp. 618-619, 577 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27963722 . Accessed: 17/07/2014 14:33 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 129.130.252.222 on Thu, 17 Jul 2014 14:33:30 PM All use subject to JSTOR Terms and Conditions

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Page 1: Some Variations on a Mathematical Card Trick

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 .

Accessed: 17/07/2014 14:33

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 ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Mathematics Teacher.

http://www.jstor.org

This content downloaded from 129.130.252.222 on Thu, 17 Jul 2014 14:33:30 PMAll use subject to JSTOR Terms and Conditions

Page 2: Some Variations on a Mathematical Card Trick

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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Page 3: Some Variations on a Mathematical Card Trick

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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Page 4: Some Variations on a Mathematical Card Trick

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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