A renowned and also atlethic rope thief arrives at a tower. Inside this tower two ropes with equal length are hanging downwards from the ceiling. The ropes are connected to the ceiling with rings. How can the thief steal as much rope as possible? The only equipment he carries is a knife.

Suppose you have two bottles with equal perimeter, thickness and height. One bottle is cilindrically shaped, the other one has the shape of a rectangular box with a square shaped bottom. In which of the bottles fits more milk?

In a suburb three new houses all have to be connected to a gas (red), water (blue) and electricity (green) supply. All houses should get their own gas, water and electricity connection. Furthermore no connections may cross each other. Because of the danger of stealing, no connections are laid below a house. In the figure on the right-hand side the situation is sketched. The red squares are the houses, the circles the point from which the connections should go to the houses. Why is it impossible to lay down all connections in such a way that no connection crosses another connection?