The following little problem makes a nice investigation for junior secondary school (and higher) learners, and was posed way back to readers in one of my Math e-Newsletters of 2006:

A bug that always crawls in a straight line, is within a circle. Whenever it reaches the edge of the circle, it turns through 135 degrees (anti-clockwise). Will the bug ever leave the circle? If so, when? If not, why not? Investigate and generalize.

Bug Escape Problem

Investigation and Explanation (proof):
1) Drag the red vertex representing the position of the bug.
2) Can you find an 'escape zone' for the bug and formulate the conditions under which it will escape the circle? Click on the provided 'Show Escape Zone' button in the sketch to check. (Note: When the bug is in the Escape Zone the correct path of the bug is given by the 'red arrow' path (followed by the 'green escape path), and not the 'black arrow' path any more. This is due to a peculiarity in the software that does not allow points on a circle to be dragged over each other.)
3) Can you explain why (prove) that the bug will escape under those conditions? Can you generalize further for turning angles other than 135 degrees?
4) Compare your solution to the following one received in 2006 from longtime mathematics friend and colleague, John Olive (now retired), from the University of Georgia, Athens, USA - click here.