How many shuffles are required to return a deck of 16 cards to their original positions if the deck is "shuffled" in the following way:
Suppose someone rearranges the numbered stickers in the first grid so that the result looks like the second grid. Find out how many such rearrangements are necessary to return the stickers to their original locations.


Construct a "multiplication table" for the symmetries of a rectangle.
Construct a multiplication table for the numbers $1, 1, i, i$.
Construct a "multiplication table" for only the rotational symmetries of a pentagon.
Define "addition modulo 5" and "multiplication modulo 5" in the following way: Define $a + b \pmod{5}$ to mean the remainder upon division by 5 of $a+b$, and define $ab \pmod{5}$ to be the remainder upon division by 5 of $ab$. Construct an addition table for numbers $0,1,2,3,4$ and a multiplication table for numbers $1,2,3,4$.
Compare the tables you have created for problems #36.
Recall that two numbers share the same remainder upon division by $n$ if and only if their difference is a multiple of $n$. With this in mind, define $a \equiv b \pmod{n}$ to mean $n \mid ba$ and prove the following are true for all integers $x$, $y$ $z$, and $n$ (with $n > 0$):