Puzzles

From Ed Burger: Suppose that $N$ cards are placed on a table with $F$ of them face up. You know the values of $N$ and $F$, but you are unable to see the cards. By moving the cards and turning them over arrange them in two piles with an equal number of cards face up.

From Frédéric Gourdeau: Find a number in base ten such that moving the last digit to the front of the number doubles its value. Formally, find a number $N$ such that: $N = d_nd_{n1}d_{n2} \dots d_3d_2d_{1}$ and $2N = d_1d_{n}d_{n1} \dots d_3d_2$. Here is a nonexample. The number $12\mathbf{3}$ is NOT such a doubling number since $\mathbf{3}12 \neq 246 = 2 \cdot 12\mathbf{3}$.

From Béla Bollobás via Nick Cheng. Show that every rational $\displaystyle 0 < \frac{p}{q} < 1$ can be expressed in the form: $$ \frac{p}{q} = \sum_{k=1}^N \frac{1}{n_k} $$ where $n_1 < n_2 < \dots < n_k$ are distinct naturals.