Problem of the Week #9: Monday Mar. 6th, 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.
A commonly used technique for shuffling a deck of playing cards goes like this:
- Split the deck down the middle into two groups containing the same number of cards each.
- Interweave the cards from the two groups by taking one card from each group, alternating until all cards are in the new deck.
Here is a video demonstration of this shuffling technique.
This is called a faro shuffle, and we are going to look at one version of it today: An out-shuffle. (If there are no interruptions, we will be looking at the other version, an in-shuffle, next week) Because 52 cards in a standard deck of playing cards become very tedious and confusing to show as an example, let’s consider a deck just of 7 cards, labeled like this:
1, 2, 3, 4, 5, 6, 7
Step one, we split the deck. In the case of an odd number of cards like this, we let the first group have the extra card:
1, 2, 3, 4 5, 6, 7
Step two, we interweave the cards. In the case of an out-shuffle, we make sure the first card in the old order stays “out of” the deck, so we start with the first group:
1, 5, 2, 6, 3, 7, 4
Now we have completed one out-shuffle on this deck of 7 cards. If we do a second out-shuffle on the deck, it would look like this:
a). How many out-shuffles in total do we need to do on a deck of 7 cards to return it to the starting order?
b). Make a chart of how many out-shuffles are needed to return each deck to its original order for decks containing 1 to 10 cards.
c). What do you notice about odd numbered decks and even numbered decks from the results in b)? Why does this happen? Will this pattern continue?
d). Extend your chart in b) to decks containing up to 20 cards.
e). What do you notice about decks whose numbers of cards are powers of 2 from the results in d)? Why does this happen? Will this pattern continue?
f). For a deck of n cards, can it ever require more than n out-shuffles to return to its original order if each shuffle is executed perfectly?
g). Share your own problem inspired by these scenarios.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.