Count the number of mixes of the card to get come back to origin

Question

Mix a deck of 52 cards by placing them in two parts and take alternate cards from both stacks. Use the cycle notation to show that it only takes 8 mixes to come back to the origin.

Since we want to come back to the origin then the permutation must be even, right? But how shall I get the cycle notation of the permutation?

I need help to start with the exercise (I prefer hints).

Answer 1

Hint number the cards 0 to 51.

• Card 0 becomes card 0.
• Card 1 becomes card 2.
• Card 2 becomes card 4.
• Card 3 becomes card 6.
• $\cdots$
• Card 26 becomes card 1.
• Card 27 becomes card 3.
• $\cdots$

Try working mod $n$ for some $n$.