Problem:
Let and be the vertices of a regular tetrahedron, each of whose edges measures meter. A bug, starting from vertex , observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let be the probability that the bug is at vertex when it has crawled exactly meters. Find the value of .
Solution:
For , let be the probability that the bug is at vertex after crawling exactly meters. Then
because the bug can be at vertex after crawling meters if and only if
it was not at A following a crawl of meters (this has probability )
and
from one of the other vertices it heads toward (this has probability ).
Now since (i.e., the bug starts at vertex ), from we have
and , leading to as the answer to the problem.
The problems on this page are the property of the MAA's American Mathematics Competitions