Andy can't measure how long he's been walking, and he can't see the curvature. He has exactly one instrument: a memory of every turn he's taken. This is the floor as he reads it.

The only difference between the two worlds

A black face is a triangle up there, and a hexagon down here

Everything else about the two surfaces is identical. White hexagon, three white neighbours, three choices each step: back, left, right. Turning the same way twice in a row means one thing in both worlds: you are walking around a black face. The only question is how many steps that takes.

The board

Four labels, forced everywhere

The tetrahedron has only four white hexagons. So at every moment, Andy's turn record pins him to one of four places: no ambiguity, no fuzziness. Replay that same record on the kitchen floor and every white tile here inherits a label: H, A, B or C, meaning this is the tile Andy would believe he's standing on.

The labels below aren't a choice I made. Start with Andy's home and propagate by turn-following, and each tile's label is forced. It closes up consistently across the whole infinite plane, which is itself the fact that makes the rest of the puzzle work.

How to read it

Discovery is a contradiction, never a suspicion

Andy is never gradually convinced. He walks, and at each tile his model reports a name. As long as it reports A, B or C, nothing can happen. He expects no pheromones and finds none, which is exactly what the tetrahedron would have given him. A five-hundred-step amble is perfectly ordinary on the sphere, so length tells him nothing, ever.

The instant his model reports H, one of two things is true. He's on the pheromones, the amble ends, and he has learned nothing. Or he's on a decoy, the tile is bare, and he knows with certainty where he is not.

Notice the asymmetry, and check it on the board: every tile carrying the real home's pheromones is labelled H. The model can produce a false H, but never a false A, B or C under his feet. Andy can be fooled in exactly one direction.

So the whole puzzle collapses into a race on this board. Andy leaves home and wanders. Sooner or later he lands on a red tile, some red tile, since he is guaranteed to hit one. If it's the one he started on, the afternoon ends quietly.

p is the probability that the first red tile he touches is not the one he left.

The walls

Every red tile is a wall, and the walls make a cage

Here is the consequence nobody expects from an infinite floor. A red tile always ends the story: either the amble is over, or Andy has already learned the truth. So for the purposes of this question, red tiles are not tiles he passes through. They are walls.

Take them off the board and see what's left. The floor doesn't stay in one piece. It shatters into separate rings of six white hexagons, each ring wrapped neatly around a single black hexagon. No ring touches another.

Andy's home sits on the corner where three of these rings meet, one per doorway. His first step picks a ring, and that is a life sentence, because the only way out of a ring is through a red tile, and stepping on a red tile is the end. He will spend his entire afternoon on six hexagons.

The board, running

Watch an afternoon go by

Everything above is a claim about a walk nobody has taken yet. Here is the walk. A real ant steps around the real kitchen floor, and a phantom ant walks the tetrahedron beside him, taking the same turn every time. The afternoon ends the first moment the phantom says H, and the only question left is whether the tile underneath is bare.

Watch the opening move in particular. Whichever ring it lands in lights up, and Andy never leaves it: not because anything stops him, but because the way out is a red tile, and a red tile is the end of the story.

What to do next

Working it out

1. Redraw a corner of the board by hand. Take home and its three neighbours, walk out three or four steps following turns, and watch the labels come out forced. Until the labelling feels inevitable rather than asserted, none of the rest will sit right.
2. Break the floor yourself. Cross out every red tile on the big board and trace what survives. You should get closed rings of six, each around one black hexagon, none of them touching. This is the step worth doing with a pencil, because it is far more convincing in your own handwriting than in mine.
3. Convince yourself the cage holds. Every tile in a ring has two neighbours along the ring and exactly one door out, and every door opens onto a red tile. So there is no route from one ring to another that doesn't end the walk first. Andy's opening move is the only choice he ever makes about which six tiles he'll die on.
4. Then solve one ring, not three. The three rings are the same ring under the symmetry about home, so the first step contributes nothing to average over. Six positions, and from each: one third to step left along the ring, one third right, one third out through that tile's door. He is safe only if he leaves by the door of the tile he arrived on.
5. Solve it exactly, then check it numerically. Six unknowns, or fewer once you use the ring's own symmetry, small enough to do on paper. A few hundred thousand simulated ambles will confirm the value to three decimals in a couple of seconds. The puzzle wants exact terms, so the simulation is never the answer; it is how you catch a sign error before you submit one.

Two brackets to aim between while you work: turning straight back on the first step ends a third of all afternoons with nothing learned, so p is at most 2/3; and the step-3 catch happens on its own 2/9 of the time, so p is at least that. When you have a number, open the panel below.