Question

If you take a deck of 2n playing cards, cut it into two stacks of n and interleave them, you have performed a perfect shuffle.

Example: for the six-card deck [1 2 3 4 5 6]^T , the result is [4 1 5 2 6 3]^T (note that the top card of the second stack goes on top. Write down the matrix A_2n for a perfect shuffle of 2n cards when 2n=10 and 2n=12. How many times must you (perfectly) shuffle a deck of 10 cards to return it to its original order? (I.e., for what k is A₁₀^k=I?) How many for 12 cards? Take a guess at the numbers for a bridge deck of 52 and a Pinochle deck of 48.

*MATLAB or any Software Language may be used. Please show all work and explain thoroughly.

Answer 1

The language used is python. I have explained each step.

import itertools #for concatenating two lists

def perfectShuffle(deck): # the perfect shuffle function will shuffle the deck of cards 'deck' one time
mid = len(deck) // 2 # finds the mid-way of the 'deck'
first_deck = deck[:mid] # divides the deck into the first half
second_deck = deck[mid:] # similarly for second half
shuffle_list = list(zip(second_deck,first_deck)) # zip() is a python function that will pair together two lists, for eg. zip([4,5,6],   [1,2,3]) = [(4, 1), (5, 2), (6, 3)]
shuffle = list(itertools.chain.from_iterable(shuffle_list)) # this function will convert the above list to a single list [4,1,5,2,6,3]
return shuffle

def countOfShuffles(start):
count = 1 # the counter will have the number of shuffles required
end = perfectShuffle(start) # the first shuffle
while (start != end): # checking whether the deck is back to original state
end = perfectShuffle(end) # if not shuffle again
count += 1
return count # return the count

def main():
A12 = list(range(1,13)) # for n = 6, 2n = 12 and A12 = [1,2,3,4,5,6,7,8,9,10,11,12]
print (countOfShuffles(A12)) #calling the function
A10 = list(range(1,11))
print (countOfShuffles(A10))
A52 = list(range(1,53))
print (countOfShuffles(A52))
A48 = list(range(1,49))
print (countOfShuffles(A48))

if __name__ == "__main__":
main()

Code

import itertools #for concatenating two lists def perfectShuffle(deck): one time # the perfect shuffle function will shuffle the deck of cards "deck mid len (deck) // 2 # finds the mid-way of the ·deck. first-deck deck( :mid] # divides the deck into the first half second-deck-deck [mid:] # similarly for second half shuffle-list list (zip(second-deck,first-deck)) # zip() is a python function that will pair together two lists, for eg. zip([4,5,6],[1,2,3])[(4, 1) (5, 2) (6, 3)] shuffle list (itert ools.chain.from-iterable(shuffle-list)) # this function will convert the above list to a single list [4,1,5,2,6,3] return shuffle def countofshuffles(start): count = 1 end - perfectShuffle(start) while (startend) # the counter will have the number of shuffles required # the first shuffle # checking whether the deck is back to original state # if not shuffle again end -perfect shuffle(end) count 1 return count # return the count

def main() A12list (range(1,13)) # for n = 6, 2n #calling the function 12 and A12 = [1,2,3,4,5,6,7, print (countOfShuffles (A12)) A10 - list (range (1,11)) print (countofShuffles (A18) A52 = list (range(1,53)) print (countofShuffles (A52)) A48 = list (range(1,49)) print (countOfShuffles (A48)) if _name- main. main " main()

Output


Python 3.6.1 (default, D ec 2015, 13: GCC 4. 8.21 on linux