II. Now suppose we consider the same questions but a slightly different sorting process. Starting with the arrangement from the previous problem:
9, 5, 6, 10, 7, 1, 4, 2, 8, 3
We start the first round the same way, from left to right, and find:
1, 2, 3
Then we do all subsequent rounds in the same direction, from left to right. The second round would give us:
4
The third round, also from left to right:
5, 6, 7, 8
The fourth round, also from left to right:
9, 10
With this new method it would take 4 rounds to sort the arrangement back to the original order.
A. There is only one arrangement that would take exactly 1 round to sort back to the original order, and that’s the original order itself:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
How many arrangements would take exactly 2 rounds to sort back to the original order?
B. Make another arrangement of these ten cards that takes 4 rounds to sort back to the original order.
C. Make another arrangement of these ten cards that gives the same sorting rounds:
1, 2, 3
4
5, 6, 7, 8
9, 10
D. How many arrangements are there that give the same sorting rounds as in Part C?
E. How do you calculate the number of arrangements that give any particular list of sorting rounds?
F. How many arrangements of these ten cards are there that would take exactly 4 rounds to sort back to the original order?
G. How many arrangements of n cards are there that would take k rounds to sort back to the original order?