Traveling Through Triangles

Relevant wiki: Expected Value - Problem Solving

Let $X_{m\rightarrow n}$ be a random variable that represents the number of moves to get from node $m$ to node $n$ .

The goal is to find $E[X_{1\rightarrow 5}]$ .

Due to the symmetry in the problem, it can be assumed that $E[X_{2\rightarrow 5}]=E[X_{3\rightarrow 5}]$ and $E[X_{4\rightarrow 5}]=E[X_{6\rightarrow 5}]$

Let $a=E[X_{1\rightarrow 5}]$ , $b=E[X_{2\rightarrow 5}]$ , and $c=E[X_{4\rightarrow 5}]$ .

Using linearity of expectation and the identities above, the following system of equations can be written:

$a=b+1$

$b=\frac{1}{4}a+\frac{1}{4}b+\frac{1}{4}c+1$

$c=\frac{1}{2}b+1$

Solving this system of equations yields $a=5$ .

Therefore, the expected number of moves required to get to node 5 is $\boxed{5}$ .