Question

let a, b, c, and d be the vertices of a regular tetrahedron each of whose edges measures 1 meter. a bug, starting from vertex a, 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 p

Answer 1

It picks one of the three edges that meet at each vertex, each of which has an equal chance of being chosen, and crawls along that edge to the vertex at its opposite end is p=182/729 and n is 182.

Let$V_k$ crawl exactly $k$ meters starting from vertex $V$ and ending at vertex A, where $V\in\{A,B,C,D\}$ and k is a positive integer. We wish to find $A_7.$

Since the bug must crawl to vertex $B,C, or D$

on the first move, we have

$where S_k=A_k+B_k+C_k+D_k.$

More generally, we get

$\[A_{k+2}=A_k+2S_k. \qquad\qquad (\spadesuit)\]$

The requested probability is

$\[p=\frac{A_7}{3^7}=\frac{2(3)+2\left(3^3\right)+2\left(3^5\right)}$

${3^7}=\frac{2(1)+2\left(3^2\right)+2\left(3^4\right)}{3^6}=\frac{182}{729},\]from which n=\boxed{182}.$

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