Problems
Here is a half of a magic trick to impress your friends. Take the Ace to Seven of hearts and order them numerically in a pile with Ace at the bottom and Seven at the top. Do the same with the Ace to Seven of spades. Put the pile of spades on top of the pile of hearts. Flip the \(14\)-card deck so the cards are now face down.
Make a cut anywhere. Deal out \(7\) cards from the top into a pile, so that you now have two piles of equal size again. Let us refer to the motion of taking the top card of a pile and putting it to the bottom as making a swap. Making two swaps means doing the motion twice, NOT taking the top two cards at the same time.
Your friend chooses two nonnegative numbers that add up to \(6\), for example \(4\) and \(2\). Make \(4\) swaps on the first pile and \(2\) swaps on the second pile. Remove the top card of each pile, so that we now have two piles of six cards.
Your friend now chooses two nonnegative numbers that add up to five and repeat the same procedure. We will continue doing this, but at each turn, the total number of swaps decreases by one. Eventually, we should have one card remaining in each pile. It turns out to be a matching pair.
How does the trick work?
Many magicians can perform what is known as the Faro shuffle. Actually there are two kinds of Faro shuffle: the in Faro shuffle and the out Faro shuffle.
Let us assume that the deck has an even number of cards. The first step of the Faro shuffle is to divide the deck into two smaller decks of equal size. One deck consist of the top half of the original deck in their original order. The other deck consists of the bottom half of the original deck in their original order.
The second step of the Faro shuffle is to interweave the two decks, so that each card is above and beneath a card of the opposite deck. This is where the in Faro and the out Faro differ: the in Faro changes the top and bottom cards of the original deck (from before step one) while the out Faro retains the original top and bottom cards.
Show that only 8 out Faro shuffles are needed to return a standard 52 card deck back into its original position.
How can you move the top card to any position in an even size deck using only Faro shuffles?
You have a deck of \(n\) distinct cards. Deal out \(k\) cards from the top one by one and put the rest of the deck on top of the \(k\) cards. What is the minimum number of times you need to repeat the action to return every card back to its position?
What are the symmetries of an isosceles triangle (which is not equilateral)?
What are the symmetries of the reduce-reuse-recycle symbol?
What are the symmetries of an equilateral triangle?
What are the symmetries of a rectangle (which is not a square)?
What are the symmetries of a rhombus (which isn’t a square)?
