As to the question of Is it always solvable - obviously not, because in the given example, you don't leave the top-left 4x4 square before running into the left wall.

(This part is not an answer)

I'd like to see what happens when we add the possibility that walking off walls moves you to the other side of the board. Then the death by walking off would not exist, but I don't know if it's always possible to reach the goal square.

EDIT - Seems to me like wrapping the sides of the board would be a little boring, because that would make the start and finish squares only 2 squares apart, and would require some playing around to guarantee you're not going to stumble into the finishing square pretty quickly.

Honestly maybe this question should just boil down to - is it possible to create a board (with wrapping edges) on which you have an infinite loop which does not walk along all of the squares.

As to the question of Is it always solvable -

obviously not,

because

in the given example, you don't leave the top-left 4x4 square before running into the left wall.

(This part is not an answer)

I'd like to see what happens when we add the possibility that

walking off walls moves you to the other side of the board.

Then

the death by walking off would not exist,

but I don't know if it's always possible to reach the goal square.

EDIT - Seems to me like wrapping the sides of the board would be a little boring, because that would make the start and finish squares only 2 squares apart, and would require some playing around to guarantee you're not going to stumble into the finishing square pretty quickly.

Honestly maybe this question should just boil down to - is it possible to create a board (with wrapping edges) on which you have an infinite loop which does not walk along all of the squares.

As to the question of Is it always solvable - obviously not, because in the given example, you don't leave the top-left 4x4 square before running into the left wall.

(This part is not an answer)

I'd like to see what happens when we add the possibility that walking off walls moves you to the other side of the board. Then the death by walking off would not exist, but I don't know if it's always possible to reach the goal square.

As to the question of Is it always solvable - obviously not, because in the given example, you don't leave the top-left 4x4 square before running into the left wall.

(This part is not an answer)

I'd like to see what happens when we add the possibility that walking off walls moves you to the other side of the board. Then the death by walking off would not exist, but I don't know if it's always possible to reach the goal square.

EDIT - Seems to me like wrapping the sides of the board would be a little boring, because that would make the start and finish squares only 2 squares apart, and would require some playing around to guarantee you're not going to stumble into the finishing square pretty quickly.

Honestly maybe this question should just boil down to - is it possible to create a board (with wrapping edges) on which you have an infinite loop which does not walk along all of the squares.