In this simulation we are experimenting with a funny jumping bug. There are piles of food at 3 points: (0,0), (1,0), and (0.5,1), and each time the bug jumps it picks one of those three piles (at random), and then jumps exactly halfway there. The bug starts at (0,0).
For example, the bug, on the first jump, might pick the pile at (0.5, 1.0), landing at (0.25, 0.5) since it only jumped halfway. Then at the next step it might randomly decide to jump towards (1,0), landing at (0.625, 0.25). Notice at each step we are averaging the x values, and also averaging the y values.

Prove the following: The bug will NEVER land on point A (0.5, 0.25).

You may want to use your line segment tools.

If the bug lands on point A, where would it have to have been jumping FROM?

Also, you may want to press the buttons several times to see all the different places that the bug might land. Does it look familiar?