You have been captured by the devil, but he proposes a game, in which if you win, you may go free. The game is as follows.
• The devil chooses a natural number k and gives you k sheets of paper.
• On every sheet, you are required to write the natural numbers from 1 to $2^k$ inclusive,
and you may write on both sides of the sheet.
• The Devil now arranges the papers side by side, choosing a side for each of them, as they like.
• The Devil wins if he can arrange them in such a way that each of the numbers from 1 to $2^k$ is on the top side of at least one of the sheets, after he has laid them down.
Who has a winning strategy?
If $k=2$, then on page 1 side 1 let us put $(2, 3)$ and on side 2 $(1,4)$.
On page 2, side 1 put $(1,3)$ and side 2 $(2,4)$.
I think this works. But can I do this for any $k$?