Once, a skater moved straight ahead for six minutes. His path was observed by several people under the following conditions:

(a) During the six minutes, someone was always watching.

(b) Everyone watched continuously for one minute.

(c) Everyone found that while they were watching, the skater moved one meter forward.

How many meters did the skater travel at least?

(A similar problem appeared in P. 159 (1972) Volume 45, Issue 10, page 222, but the solution of the practice problem does not require knowledge of the solution to that problem.)
If the speed of the robot is not uniform, then the estimates of its path can also be influenced by the number and arrangement of observers. If six people are observing, then they can only organize their work in one way. The first starts observing at the start, and then they take turns every minute. In this case, the robot clearly travels a total of six meters. However, with more than six observers, there is a period when more than one person is observing the robot, and the robot can reduce the distance it travels if it moves during these times.

![](https://cdn.mathpix.com/cropped/2024_05_02_db56ad9061a136c5b729g-1.jpg?height=271&width=1391&top_left_y=390&top_left_x=367)

With seven observers, we can already arrange them so that no one observes the robot's path alone. The seventh person can enable a total of sixty seconds of simultaneous observation. If we divide this, for example, into six equal parts, meaning the subsequent observers start their observation every 50 seconds after the start, then the last one will finish observing exactly at the end of the 6th minute.

The robot can "handle" the first six observers in pairs if it moves forward one meter in each of the three ten-second intervals when the first and second, third and fourth, and fifth and sixth persons observe it simultaneously. If it then moves another meter in the last ten seconds, all conditions are met, and the robot's path is only four meters.

We will show that the robot cannot travel less than this, regardless of the number of observers and how they organize their observation under the given conditions.

We know that the robot was observed continuously for six minutes. Therefore, it was observed at the start, and also at the end of the 2nd, 4th, and 6th minutes. Since the difference between these time points is exactly two minutes, and each person observes for one minute, the robot was definitely observed by four different people at these four time points. Each of these four saw the robot travel one meter. Regardless of when the robot moved during any of their observation periods, this distance could not have been detected by the other three, as no two of the four observers had overlapping observation times. This means that the robot must have traveled four meters during the observation periods of these four people.

Note. Some interpreted the conditions to mean that the same person could observe the robot for one minute multiple times with interruptions. In this case, the problem is much simpler. Since there is an observer, the robot must travel at least one meter. On the other hand, if two people take turns, and if they observe simultaneously for the entire time, the robot can just travel this one meter during this period.