(5.13) How many shuffles? (Mathematics) (3 How many shuffles does it take to randomize a deck of 52 cards? The answer is a bit controversial; it depends on how one measures the information left in the cards. Some suggest that seven shuffles are needed; others say that six are enough. $2 We will follow reference [141], and measure the growing randomness using the information entropy. We imagine the deck starts out in a known order (say, AA, 20, . .., K&). (a) What is the information entropy of the deck before it is shuffled? After it is completely ran- domized? The mathematical definition of a riffle shuffle is easiest to express if we look at it backward. 53 Consider the deck after a riffle; each card in the deck either came from the upper half or the lower half of the original deck. A riffle shuffle makes each of the 232 choices (which card came from which half) equally likely. (b) Ignoring the possibility that two different rif- fles could yield the same final sequence of cards, what is the information entropy after one riffle? You can convince yourself that the only way two riffles can yield the same sequence is if all the cards in the bottom. half are dropped first, fol- lowed by all the cards in the top half. (c) How many of the 232 possible riffles drop the entire bottom half and then the entire top half, leaving the card ordering unchanged? Hence, what is the actual information entropy after one riffle shuffle? (Hint: Calculate the shift from your answer for part (b).) We can put a lower bound on the number of riffles needed to destroy all information by as- suming the entropy increase stays constant for future shuffles. (d) Continuing to ignore the possibility that two different sets of m riffles could yield the same fi- nal sequence of cards, how many riffles would it take for the entropy to pass that of a completely randomized deck