Question 111
Let $n \geq 2$ be a natural number. In the middle of a circular garden stands a watchtower. At the edge of the garden stand $n$ equally spaced garden gnomes. Attentive guards live on the watchtower. Each guard watches over a region of the garden which is bounded by two different garden gnomes.
We say that guard $A$ controls guard $B$ if the whole region of $B$ is contained in that of $A$.
Among the guards there are two groups: apprentices and masters. Each apprentice is controlled by exactly one master and himself controls nobody, whereas masters are controlled by nobody.
The whole garden has maintenance costs:
- An apprentice costs 1 gold piece per year.
- A master costs 2 gold pieces per year.
- A garden gnome costs 2 gold pieces per year.
Show that the garden gnomes cost at least as much as the guards.
We say that guard $A$ controls guard $B$ if the whole region of $B$ is contained in that of $A$.
Among the guards there are two groups: apprentices and masters. Each apprentice is controlled by exactly one master and himself controls nobody, whereas masters are controlled by nobody.
The whole garden has maintenance costs:
- An apprentice costs 1 gold piece per year.
- A master costs 2 gold pieces per year.
- A garden gnome costs 2 gold pieces per year.
Show that the garden gnomes cost at least as much as the guards.