Question 34
A deck of $2n+1$ cards consists of a joker and, for each number between $1$ and $n$ inclusive, two cards marked with that number. The $2n+1$ cards are placed in a row, with the joker in the middle. For each $k$ with $1 \leq k \leq n$, the two cards numbered $k$ have exactly $k-1$ cards between them. Determine all the values of $n$ not exceeding $10$ for which this arrangement is possible. For which values of $n$ is it impossible?