Three crowds leave home at the same moment, each walker taking one random step after another. The first crowd can only go up or down a street. The second is loose in a city grid. The third can fly: six directions, open sky. A walker turns gold the first time it stumbles back through its own front door. Watch which crowds turn gold, and which one stops.
For the curious kid.
Imagine flipping a coin for every step: heads forward, tails back. You wobble away from your door, and wobble back, and it feels like you might drift off for good. You will not. On a street, every walker comes home, always. In a city, with four ways to turn, it is still true, though it can take an astonishingly long time. But give the walker wings and it breaks. In the sky only about one in three ever sees home again. The rest are gone for good, and no amount of waiting brings them back.
Why? Count the room. After a thousand steps a street walker has drifted only about thirty doors from home, so it has trodden those few doors over and over, its own included. A city walker has about a thousand corners within reach and a thousand steps to spend: just barely enough to keep finding the old ones. A bird has about thirty thousand places within reach and the same thousand steps. Most of the sky it will never touch even once, and one of the places it may never touch again is home.
Deeper.
George Pólya proved it in 1921: the simple random walk on a lattice returns to its start with certainty in one and two dimensions, and with probability less than one in three or more. Shizuo Kakutani gave it the line everyone remembers: "A drunk man will find his way home, but a drunk bird may get lost forever."
The argument is a piece of bookkeeping. Let u(n) be the chance the walker is home at step n. If the chance of ever returning is p, then each homecoming is a fresh start, the number of visits home is geometric, and its average is 1/(1 − p). So p = 1 exactly when the sum of all the u(n) is infinite. On the street u(2n) = C(2n, n)/4ⁿ, which shrinks like 1/√(πn): the sum diverges. In the city, turn the map 45 degrees and the two diagonal coordinates are two independent street walks, so u is the street's value squared, about 1/(πn): the sum still diverges, but only like a logarithm, the slowest way a sum can. In the sky u shrinks like n to the power −3/2 and the sum converges to 1.5164, which gives p = 1 − 1/1.5164 = 0.3405. That number has an exact form in gamma functions (Watson 1939; Glasser and Zucker 1977), and the page computes it from that.
The gold lines on the chart are not simulations. They are the exact chance of having been home by step n, obtained from u by peeling off first returns one at a time. By 100 steps: 92.0% on the street, 58.3% in the city, 31.2% in the sky. By 10,000: 99.2%, 73.9% and 33.8%, with the sky pressed against its ceiling of 34.05%. The city's line does reach everyone, but the leftover fraction falls only like π divided by the logarithm of the time: for nine walkers in ten to be home you would wait on the order of a trillion steps. And a quiet shock sits in the first two worlds: return is certain, yet the average wait to come home is infinite.
Honesty notes, museum policy. Walkers move one lattice step at a time along an axis, independently, by pseudo-random choice. Each panel zooms out as the crowd spreads (as the square root of time), so the clouds look the same size while really growing a hundredfold. The sky is drawn slowly turning so you can read its depth. The laws were checked headless against 20,000 walkers per world at four times, all within statistical error, and the sky's limit reproduced Pólya's number to seven digits. With 2,500 walkers a crowd, a measured curve may wander about a percentage point from its line; release more crowds and it tightens. Somewhere past the sky there is one more world, named for the man who found the rule.
Getting lost is not about how far you wander. It is about how much room there is to never cross your own path.