Question 50
Consider a stack of cards numbered $1$ to $104$. A shuffle interleaves the top $52$ cards with the bottom $52$ so that the former top card becomes second from the top. The figure shows two successive shuffles from the initial order, increasing from top to bottom.
a) Complete the table.
Number of shuffles $k$ / position $p_k$ of card number $5$, counted from the top:
$$\begin{array}{c|rrrrrr}k&1&2&3&4&5&6\\p_k&10&?&?&?&?&?\end{array}$$
b) From the initial position, where does card $n$ go after one shuffle?
c) Find two cards that exchange positions with each other at every shuffle.
d) A group of three cards that exchange positions among themselves at each shuffle is called an invariant trio. Find all invariant trios starting from the initial position.
a) Complete the table.
Number of shuffles $k$ / position $p_k$ of card number $5$, counted from the top:
$$\begin{array}{c|rrrrrr}k&1&2&3&4&5&6\\p_k&10&?&?&?&?&?\end{array}$$
b) From the initial position, where does card $n$ go after one shuffle?
c) Find two cards that exchange positions with each other at every shuffle.
d) A group of three cards that exchange positions among themselves at each shuffle is called an invariant trio. Find all invariant trios starting from the initial position.