Problems
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)?
There are six symmetries of an equilateral triangle: three reflections, and three rotations (thinking of the identity as one the rotations). Label the three reflections \(s_1\), \(s_2\) and \(s_3\). Label the identity by \(e\), rotation by \(120^{\circ}\) as \(r_1\), and rotation by \(240^{\circ}\) clockwise as \(r_2\).
Note the following definition: Each symmetry has an inverse. Suppose we apply symmetry \(x\). Then there is some symmetry we can apply after \(x\), which means that overall, we’ve applied the identity. What are the inverses of \(r_1\) and \(s_1\)?