 Solution

After the first of the four possible steps, a self-avoiding walk has at most three choices at every stage because it cannot immediately reverse its preceding step. Thus $b_n\le 4\cdot 3^{n-1}$ and $\kappa \le 3$. On the other hand, every sequence of $n$ north or east steps is self-avoiding, so $b_n\ge 2^n$ and $\kappa \ge 2$.