The Ant on a Cube

The problem
random-walk along cube edgesstartopposite cornerE[steps] = ?

An ant sits at one corner of a unit cube. Every second it walks along one of the three edges meeting its current corner, chosen uniformly at random.

On average, how many steps does it take to reach the corner diagonally opposite from its start?

Tempting (but wrong)
"3 or 4 steps?"shortest path is 3 — actual expectation way higher

Most people anchor on the shortest path: 3 edges. So they guess 3, 4, or maybe up to 6 steps.

But the ant isn't routing — it's walking randomly. At every corner, only 13\tfrac{1}{3} of its choices push it closer; 23\tfrac{2}{3} of choices keep it the same distance or move it back. The expected time is much larger than the diameter.