Need B, C & D. other than a square that has exactly eight symmetries, and use yl
ID: 3283676 • Letter: N
Question
Need B, C & D.other than a square that has exactly eight symmetries, and use yley table of that object's symmetry group to show that it is not equivalent to D. Find at least one object of to Buffalo. The inshuffle and outshuffe are two varieties of perfect shuffles. e, a deck with an even number of cards is split exactly in two, and the two halves are then interwoven so that every other card in the shuffled deck comes from the same half. The outshuffle returns the top card to the top of the deck and the bottom card Shuffle For each shuffl to the bottom. The inshuffe returns the top card to the second position from the top. a) (Test - don't include in your writeup) For a deck of six cards, write the inshuffle and the outshuffle in cycle notation. Verify that the inshuffle has order 3 and the outshuffle has order 4 b) Find the order of the inshuffle and the outshuffle in a deck of 52 cards. Use the con- struction G-SymmetricGroup (52), as illustrated in the "Permutation Groups" section of Beezer's notes. e) If an ambidextrous mathemagician alternates between inshuffles and outshuffles, will the deck eventually return to its original order? (Show a calculation that supports your conclusion. ) What about imperfect shuffles? If you mess up the deck in the same way over and over again, must it eventually return to its original order? What is the largest order you can find of a shuffle (permutation) of 52 cards? Crazy Eights. Further investigate the groups of order 8 by exploring groups of units nodulo n. Is there any n such that Un is essentially Q in disguise? ($7.1, exercise 16) What about the whackadoodle group from exercise 36? What other groups of order 8 can ou discover? Show Cayley tables for each different group, and explain how you know they re not just re-labellings of one another. fact, these two permutations are sufficient to construct the entire symmetry group of the triangle, muc three rotations are given in the cube example.
Explanation / Answer
b)
Assume,
have N shuffled decks. then probability that no 2 are the same is
1 - (1-(1/52!))^(N/2).
if we want this probability to be around 50, then N will be around 10^34.
Let's assume a shuffle per second. this will take about, around 10^16 age of the universe or 10^26 years.
Now instead of 1 shuffle per second, let's take 1 shuffle per quantum transmission. this event takes approximately 10^-10 seconds.
Even if we shuffle so fast we will still have to wait for 10^6 ages of the universe. However, if we every human on earth shuffled at such a speed it would take 4.5 million years.
For example,
the first fossil records of mammoths date to 5 million years ago and the first records of Australopythecus are 4 million years old.
Using an out-shuffle, a deck originally arranged as 1 2 3 4 5 6 7 8 would become 1 5 2 6 3 7 4 8. The ordering of a deck of 52 cards after an out-shuffle is given by 1, 27, 2, 28, 3, 29,...
Algorithm to bring a card at position p to position 0
Working with the deck of 2n cards, define r by 2^(r-1)<2n<2^r
(so if 2n = 52 then r=6).
For 0<p<2n-1, let t=[((p+1)2^r)/2n]. For p=0, set t=0: for p=2n-1, set t=2^r-1.
Express t in binary as t=t(r-1)t(r-2)...t1t0(with ti=1 or 0). Define "correction terms" s'=2nt-(2^r)p=s(r-1)s(r-2)...s1s0(with si=1 or 0).
The shuffling sequence is t(r-1)+s(r-1),t(r-2)+s(r-2),......t0+s0, where each sum is in binary with 1 as in and 0 as out. Any trailing 0's can be deleted.
c)
It depends on mathematical model. There are 52! permutations of a standard deck of playing cards without jokers, which is on the oder of 10^67. Perfect shuffles are apparently are called Faro shuffle, in which cards are separated into 2 poles of 26 and interlaced one over top if the other. 8 such shuffles should retrain the order of the deck. Using an out-shuffle, a deck originally arranged as 1 2 3 4 5 6 7 8 would become 1 5 2 6 3 7 4 8. The ordering of a deck of 52 cards after an out-shuffle is given by 1, 27, 2, 28, 3, 29,...
A perfect shuffle leaves the top card the same. after the 1st shuffle say queen of hearts is moved to the 3rd position. after the 2nd shuffle, queen is moved to the 5th position. for the next shuffle the queen moves to the 9th position.
If it has position k, it will move to position 2(k-1), as you set cards on top of it.
d)
If one manages to perform 8 perfect faro out-shuffles in a row, then the deck of 52 cards will be restored to its original order. If one can do perfect in-shuffles, then 26 shuffles will reverse the order of the deck and 26 morewill restore it to its original order.
About 7 for a deck of 52 cards, times have to shuffle a deck of cards in order to mx them reasonably well.
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