Ant on a tetrahedron, reaching the far vertex

Question

A tetrahedron has four vertices, and every vertex is joined to the other three. An ant starts at vertex A. Each minute it crawls along an edge to one of the three neighboring vertices, chosen at random. Vertex D is the target.

On average, how many minutes until the ant first reaches vertex D?

Accepted Answer

Let a be the expected time from the start A, and b the expected time from either of the other two non-target vertices (B and C, equal by symmetry). From A the ant reaches D with chance 1/3 (done) or moves to B or C: a = 1 + 	frac13 \cdot 0 + 	frac23 b. From a non-target vertex like B, its three neighbors are A, the other non-target C, and target D: b = 1 + 	frac13 a + 	frac13 b + 	frac13 \cdot 0 \implies 	frac23 b = 1 + 	frac13 a \implies b = 	frac32 + 	frac a2. Substitute into the first equation: a = 1 + 	frac23\left(	frac32 + 	frac a2ight) = 1 + 1 + 	frac a3 = 2 + 	frac a3 \…

Ant on a tetrahedron, reaching the far vertex

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