A FLY'S VIEW - MORE TOPOLOGY PUZZLES

A FLY'S VIEW

A topologist views geometrical figures a little differently than an ordinary mathematician. 

For example, he recognizes that the out-side of a figure is just as important as the inside, and that the kind of movement possible between outside and inside reveals some-thing important about the mathematical nature of the figure. 

Also, since a topologist loves to change one figure into another, he knows that a figure's inside and outside may exchange places and become an interesting variant in understanding transformations.
These two puzzles show a simple way to become more aware of outside and inside.

A fly lands inside each of the shapes below and tries to cross each side only once in order to wind up outside the shape. You can see that it has no trouble doing this with the triangle.

 On which of the other shapes can he do this ?

This time, the fly wants to begin and end inside the shape, crossing each side only once. You can see that he can't do it with the triangle because crossing the third side takes him outside the shape. 

Based on what you've concluded from the first puzzle, what sorts of shape do you think the fly needs ?

This maze makes inside and outside a little harder to tell apart. 

Can you tell at a glance whether each number points to the inside (enclosed with walls) or outside (open corridors or atriums) of the maze ? 

After wandering in and out of simple geometrical shapes all day, the fly might know. . . .