Task 3. (Option 1)
A cleaning robot moves at a constant speed and is programmed to turn 90 degrees every 15 seconds, and moves in a straight line between turns. Can we expect the robot to return to the starting point after 6 minutes?
Problem 676
Official solution
Solution: The number of horizontal segments along which the robot moves for 15 seconds, moving away from the initial position, is equal to the number of horizontal segments along which it moves, approaching the initial position (this is easy to see if you draw a movement diagram that satisfies the conditions). Therefore, the number of horizontal segments is even. Similarly, the number of vertical segments is even. Moreover, any horizontal movement corresponds to one vertical movement, meaning there are an equal number of horizontal and vertical segments, and the total number of movement segments is divisible by 4. 15 sec sec. Therefore, the robot's return to the initial position can be expected at any whole number of minutes.
Answer: the robot's return to the initial position can be expected at any whole number of minutes