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: $N = d_nd_{n-1}d_{n-2} \dots d_3d_2d_{1}$ and $2N = d_1d_{n}d_{n-1} \dots d_3d_2$. Here is a non-example. 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.