Suppose that we want to patrol a fence (line segment) using k mobile agents with given speeds v_1, ..., v_k so that every point on the fence is visited by an agent at least once in every unit time period. A simple strategy where the i-th agent moves back and forth in a segment of length v_i/2 patrols the length (v_1 + ... + v_k)/2, but it has been shown recently that this is not always optimal. Thus a natural question is to determine the smallest c such that a fence of length c(v_1 + ... + v_k)/2 cannot be patrolled. We give an example showing c >= 4/3 (and conjecture that this is the best possible). We also consider a variant of this problem where we want to patrol a circle and the agents can move only clockwise. We can patrol a circle of perimeter r v_r by a simple strategy where the r fastest agents move at the same speed. We give an example where we can achieve the perimeter of 1.05 max_r r v_r (and conjecture that this constant can be arbitrary big). We propose another variant where we want to patrol a single point under the constraint that each agent i = 1, ..., k can visit the point only at a predefined interval of a_i or longer. This problem can be reduced to the discretized version where the a_i are integers and the goal is to visit the point at every integer time. It is easy to see that this discretized patrolling is impossible if 1/a_1 + ... + 1/a_k < 1, and that there is a simple strategy if 1/a_1 + ... + 1/a_k >= 2. Thus we are interested in the smallest c such that patrolling is always possible if 1/a_1 + ... + 1/a_k >= c. We prove that alpha <= c < 1.546, where alpha = 1.264... (and conjecture that c = alpha). We also discuss the computational complexity of related problems.