One way to shuffle playing cards is to use

a perfect shuffle. First, you divide a deck of 52 cards into two halves of 26

cards each. Next, you merge the halves by interleaving the cards as follows.

Beginning with the top half and alternating halves, you take the bottom card

from a half and place it on top of a new deck. For example, if our deck

contains the six cards 1 2 3 4 5 6, the top half is 1 2 3, and the bottom half

is 4 5 6. The 3 at the bottom of the top half becomes the bottom card in the

shuffled deck. We then place the 6, which is at the bottom of the bottom half,

on top of the

One way to shuffle playing cards is to use

a perfect shuffle. First, you divide a deck of 52 cards into two halves of 26

cards each. Next, you merge the halves by interleaving the cards as follows.

Beginning with the top half and alternating halves, you take the bottom card

from a half and place it on top of a new deck. For example, if our deck

contains the six cards 1 2 3 4 5 6, the top half is 1 2 3, and the bottom half

is 4 5 6. The 3 at the bottom of the top half becomes the bottom card in the

shuffled deck. We then place the 6, which is at the bottom of the bottom half,

on top of the shuffled deck. Next, we place 2 on top, then 5, 1, and finally 4.

The shuffled deck is then 4 1 5 2 6 3. Notice that the card that was on top of

the original deck is now second in the shuffled result, and the bottom card in

the original deck is now second from the bottom in the shuffled deck. This

shuffle is called an in-shuffle and is achieved by beginning with the top half

when you move cards into the shuffled result. If you begin with the bottom

half, you get an out-shuffle, whereby the original top card and bottom card

remain in their positions in the shuffled deck. Define a class of playing-card

decks by using a deque to contain the cards. Your class should define methods

to perform perfect in-shuffles and perfect out-shuffles. Using your class,

a. How many perfect out-shuffles are needed

to return a deck of n cards to its original order?

b. How many perfect in-shuffles are needed

to return a deck of n cards to its original order?

c. You can move a decks top card, which is

at position 0, to any desired position m by performing a sequence of

in-shuffles and out-shuffles, as follows. You write m in binary. Beginning with

the leftmost 1 and proceeding to the right, you perform an in-shuffle for each

1 encountered and an out-shuffle for each 0. For example, if m is 8, we have

1000 for its binary equivalent. We would perform one in-shuffle followed by

three out-shuffles to move the original top card to position 8, that is, so it

is the ninth card from the top of the deck. Define a method to perform this

card trick.

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