= Solution
States $i,j$ communicate when each is accessible from the other: $(P^m)_{ij}>0$ and $(P^n)_{ji}>0$ for some nonnegative integers $m,n$. The communicating classes are
$$
\boxed{\{1,4\}\text{ closed},\qquad\{2,3,5\}\text{ open},\qquad\{6\}\text{ closed}.}
$$
Indeed $1$ and $4$ move only between themselves; $2\to5\to3\to2$ links the second class, but it has exits to $1,4,6$ and no return from the closed classes; state 6 is absorbing.
Let $h_i$ be the <probability> of ever reaching 6 from state $i$. Then $h_1=h_4=0$, $h_6=1$, and first-step conditioning gives
$$
h_2=\frac34h_5,\qquad h_3=\frac12h_2+\frac12h_3,\qquad h_5=\frac14h_3+\frac14h_4+\frac14h_5+\frac14.
$$
Thus $h_3=h_2=3h_5/4$ and $3h_5=h_3+1$, giving $h_5=4/9$. Therefore
$$
\boxed{\mathbb P_3(\text{eventual absorption in }6)=\frac13.}
$$
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