Placing a Deck of Cards in Order

Placing a Deck of Cards in OrderAuthor(s): Mark YatesSource: The Mathematics Teacher, Vol. 98, No. 5 (DECEMBER 2004/JANUARY 2005), pp. 312-315Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27971715 .

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Placing

a Deck

of Cards

in

Order Mark Yates

While playing cards one evening with

my family, I became curious about whether I was wasting moves while

reordering my cards. After each hand was dealt, I studied it for a

minute until I thought that I had determined the

smallest number of moves needed to arrange the

cards by suit. After a while, I wondered whether I

could approach this topic mathematically. The question: What is the minimum number of

moves necessary to arrange a "maximally disordered" deck of cards by suit (not necessarily in numerical

order)? other words, I wanted to place all

the hearts together, all the spades, all the

clubs, and all the diamonds. Before read

ing further, try to find the answer on

your own. Play around with actual cards in your hands to make

solving this problem easier.

Try solving a simpler

problem, using fewer

suits. Organize your data.

Look for patterns while you build in complexity. Addition

ally, can you find an equation that *M??. *J?* produces the pattern?

In this article, the hearts are abbre

viated H; clubs, as C; diamonds, D; and



spades, S. The first question to be considered is, What is the meaning of maximally disordered? Ex

amine the simplest case first, that of only H and C.

Consider the pattern HHCHC. It takes only one

move to place these cards in the desired order.

What is the definition of a move? A move consists

of taking a single card from one location in your hand and putting it in another location. It is not an

interchange. An interchange means swapping the lo

cation of two cards and is considered two moves.

Another pattern with an equal number of cards

is HCHCH. It takes two moves to place them in

order, so the pattern is more disordered than the

previous pattern. The sequence of pictures in figure 1 shows ar

ranging eight cards in three moves.

Another example can be created by including an

other suit to obtain the pattern HCHDCDD, which can

be arranged in two moves. Move the second C next to

the first; then move the second H next to the first. The same set of cards in the order DCHDCHD takes three

moves. After similar trials, you can see that different

arrangements of the same cards need different moves to

order the suits. When the cards are arranged in a pat tern that repeats, for example, HDCSHDCSHDCS,

oddly enough, a maximum disorder exists. A good way to introduce this problem to students

and to clarify the meaning of maximum disorder is to

do the following. Have the students pair up, and give four cards of each of two suits to each group. The first student shuffles the cards randomly and hands them to

the other student to arrange by suits. They record the number of moves needed to arrange the eight cards.

They reverse roles and do it again. The teacher polls the class to learn how many moves are needed and then poses a challenge. Ask one person in each group to place the cards in an order that forces the student's

partner to take the maximum number of moves to sort

the cards. The students then reverse roles and repeat the process. After the students agree that three moves are needed to arrange the hand, you can have them

progressively add cards to the two suits until they have thirteen of both suits. They can then add more suits.

Encourage the students to organize their data. Be

ginning with just a few cards, students can enter into a chart the number of moves necessary to arrange the cards. As they increase the number of cards, they continue entering the number of moves. They can then look for patterns in the data. Students can com

plete a chart similar to the one shown in figure 2. Maximum disorder is loosely defined as the

arrangement of cards for which the greatest num ber of moves is needed to arrange the cards. For a

given number of cards or suits, some arrangements take the same number of moves to order. For exam

ple, HCCH and HCHC both take one move, so they are equally disordered. Fig. 1 Arranging eight cards in three moves

Vol. 98, No. 5 ? December 2004/January 2005 | MATHEMATICS TEACHER 313



TABLE 1

F?q. 2 Chart used for recording number of moves needed to arrange cards by suit

The original problem can be restated in simpler terms: If the fifty-two cards in a deck are arranged in a

repeating pattern so that every heart, spade, diamond, and club has three other suits between it and the next

card of the same suit, then what is the minimum num

ber of moves needed to arrange the deck by suit?

First, solve a simpler problem: What is the mini mum number of moves required to arrange by suit the following cards: HCHCHCHCHC? Complete a

chart, such as the second column of figure 2. Note that with two cards, zero moves are needed;

with three cards (two cards of one suit and one card of the second suit), one move is needed; with five or

six cards, two moves are needed; and so on.

The term ceiling function must be defined before

giving a formula for two suits. A ceiling function, de noted as [x\ takes a decimal and rounds it up to the least integer greater than or equal to the decimal. For

example, 10/91 = 2. The ceiling function is similar to

the familiar greatest integer function, which rounds down to the greatest integer less than or equal to the

decimal, but calculators have no button for the ceil

ing function. Returning to the case of two suits, if m

represents the number of moves and represents the number of cards, then the formula is

m = -1.

It is a step function.

Next, solve a slightly harder problem: What is the minimum number of moves required to arrange

by suit the following cards: HCSHCSHCSHCSHCS? Set up a chart, and look for a pattern.

With three cards, zero moves are needed; with

four, one move is needed; with five or six, two moves are needed; with seven, three are needed; with eight or nine, four are needed; and so on. The formula for three suits is

m = In -2.

Finally, solve the following problem: What is the minimum number of moves required to arrange by suit the following cards: HCSDHCSDHCSDHCSD

HCSD? Set up a chart, as shown in table 1, and look for a pattern.

Note that with four cards, zero moves are needed; with five, one move is needed; with six, two moves are needed; with seven or eight, three moves are

needed; and so on. The formula for four suits is

m = 3n -3.

For = 52, the number of moves needed is thirty-six. Recognition of the pattern happens before the

formulas are invented. Students notice that for two

suits, the pattern is two l's, two 2's, two 3's, and so on. For three suits, the pattern is one 1, two 2's, one 3, two 4's, one 5, two 6's, and so on. Finally, for four suits, the pattern is one 1, one 2, two 3's, one

4, one 5, two 6's, one 7, one 8, two 9's, and so on.

Most of my students just ran this pattern to fifty two for the final answer.

Another way to solve this problem: Picture your self holding the entire set of fifty-two cards (no mean

feat) in the requisite order DHCSDHCSDHCS.... Twelve moves are needed to get the D's together, twelve to get the C's together, and twelve to get the

H's together. The S's are then automatically together. The result is the aforementioned thirty-six moves.

As an extension to this problem, a move can be

redefined. What if a move allows you to move a

group of adjacent cards at one time? For example, if

your cards were HDHCCHHHC, then you could

take the two C's that are together and move them to

the far right. Then you would move the D to the far

left. The ordering can then be completed in only two moves. Given the new definition of a move,

314 MATHEMATICS TEACHER | Vol. 98, No. 5 ? December 2004/January 2005



what would be considered maximally disordered? How many moves are

needed to arrange the cards? If thirty-six moves are needed to

order the cards, is it true that thirty-six moves are need to maximally disorder them? There is more than one way to

maximally disorder a given number of cards. For example, DSCHDSCH is as

disordered as DSCHDCSH. How many ways exist to get a maximum disorder with two suits, three suits, or four suits?

A special thanks to one of the Mathe matics Teacher reviewers, who discov ered the formulas given in this article, oo

MARK YAT?S, myate$@

Ne is ?rt mod

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Placing a Deck of Cards in Order

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