shows that $n_b/L$ goes to zero in the limit $L\rightarrow\infty$ in the region below $p=0.5$ and furthermore that this size scaling is a power law $n_b/L\propto L^{-\alpha}$. Figure \[fig:frac\](b) shows that the power-law index decreases with increasing $p$ and vanishes at a critical value $p_{c2}\approx 0.5$. Since this vanishing slope is clearly in itself a critical behavior, it is plausible that the slope vanishes at the exact threshold $p_{c2}\leq 0.5$. Again the formation of a single percolating cluster at $p_{c2}$ is likely to be associated with a sudden change of the properties of the routes which take you from the bottom to the top. Since a sudden change in the number of backsteps is just such a property, this suggests that $\alpha$ vanishes precisely at $p_{c2}=0.5$. The numerical results are consistent with vanishing at $p=0.5$, but our numerical precision is not high enough to rule out that it could also vanish somewhere in the region $0.49<p<0.5$.

The fact that the minimum number of backsteps per level vanishes at least below $p=0.49$ makes it possible to deduce that the tagged-route value for the lower threshold $p_{c1}=\frac{1}{2}\sqrt{13}-\frac{3}{2}\approx 0.30278$ is exact. The reason is that a typical path from the bottom to the 