MathDB

Fiona has a deck of cards labelled

1

n

, laid out in a row on the table in order from

1

n

from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:

[*]If at some stage the cards are in

m

piles, she chooses

1\leq k<m

and arranges the cards into

k

piles by picking up pile

k+1

and putting it on pile

1

; picking up pile

k+2

and putting it on pile

2

; and so on, working from left to right and cycling back through as necessary.
She repeats the process until the cards are in a single pile, and then stops. So for example, if

n=7

and she chooses

k=3

at the first step she would have the following three piles:

\begin{matrix} 7 & \ &\ \\ 4 & 5 & 6 \\ 1 &2 & 3 \\ \hline \end{matrix}

If she then chooses

k=1

at the second stop, she finishes with the cards in a single pile with cards ordered

6352741

from top to bottom.
How many different final piles can Fiona end up with?

Problem Statement