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 $b_n\leq4\cdot3^{n-1}$ and $\kappa\leq3$. On the other hand, every sequence of $n$ north or east steps is self-avoiding, so $b_n\geq2^n$ and $\kappa\geq2$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 204 / 1 / b / Solution

 Solution (source code)