The drunk walker
A walker starts at home and takes random steps: north, south, east or west on a flat grid, plus up or down in 3D. Will it ever come back home?
One walker on a flat grid. The orange dot is home, and the trail fades as it goes.
Now release 3,000 walkers in each world and count how many have made it home at least once.
0%flat grid (2D) home at least once
0%cube (3D) home at least once
0steps each
Both curves climb quickly at first. Then the 3D curve stalls at around 34% and stays there no matter how long you wait. The 2D curve slows down too, but it never stops climbing. Given enough time, every walker on a flat grid gets home.
In two dimensions there's nowhere to get lost, but add one more dimension and two-thirds of the walkers wander off forever.