Problem of the Week #10: Monday Mar. 13th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
This is the continuation of last week’s problem about an out-shuffle.
Now let’s consider the other variation of a faro shuffle called an in-shuffle, also with the example of a deck containing 7 cards:
1, 2, 3, 4, 5, 6, 7
Here we also do the same two steps, except we make all the opposite choices in the details.
Step one, we split the deck. In the case of an odd number of cards, we let the second group have the extra card:
1, 2, 3 4, 5, 6, 7
Step two, we interweave the cards. For an in-shuffle, we make sure the first card in the original order is shuffled “into” the deck, so we start with the second group:
4, 1, 5, 2, 6, 3, 7
If we do another iteration of in-shuffle, it would look like this:
a). How many in-shuffles in total will it take to return a deck of 7 cards to its original order?
b). Make a chart of how many in-shuffles are needed to return each deck to its original order for decks containing 1 to 10 cards.
c). Compare the chart in b) to the chart you produced for the out-shuffle portion of this problem posted last week. What do you notice? Why does this happen? Will this pattern continue?
d). For a deck of n cards, can it ever require more than n in-shuffles to return to its original order if each shuffle is executed perfectly?
e). If we mix the two variations, what sequences of out-shuffles and in-shuffles can return a deck to its original order?
f). Share your own problem inspired by these scenarios.
g). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.