Brain Teaser????

[Scotty O]

A friend an I were talking about shuffling chips. He brought up a cool observation:

2 stacks of 2, 2 Reds 2 Green. Shuffle them perfectly with your hands and in 2 turns you will have 2 of the same colored stack.

Now 3 and 3, it takes 3 turns of a perfect shuffle to get the full colors back

Now 4 and 4, it takes 3 times.

Now 5 and 5, it seems you can never get them back? or Can you?

Just a visual:

G R
G R
shuffle to make
G
R
G
R
Split the pile
G G
R R
Shuffle
G
G
R
R
Split
G R
G R

Thoughts for math wiz's out there.

[gaming_mouse]

After 10 shuffles, all chips will return to their original positions. Starting with a single pile, which you will then break into two piles and shuffle together, the position mapping is as follows (1=bottom chip):

1-->2
2-->4
3-->6
4-->8
5-->10
6-->1
7-->3
8-->5
9-->7
10-->9

Follow the path of 1, eg:

1-->2-->4-->8-->5-->10-->9-->7-->3-->6-->1

The other chips are similar.

By some theorem of group theory or other, this will always be the case, no matter how big the piles. That is, eventually you get what you started with.

gm