Project Euler Problem 607
Frodo and Sam need to travel 100 leagues due East from point A to point B.
Solution
Answer: 13.1265108586
Let the marsh boundaries be parallel lines running South-West to North-East.
The optimal path obeys the continuous analogue of Snell’s law:
$$\frac{\sin\theta_i}{v_i}=k$$
where:
- $v_i$ is the speed in region $i$,
- $\theta_i$ is the angle to the normal of the marsh boundaries,
- $k$ is a constant.
The seven regions have speeds
$$(10,9,8,7,6,5,10)$$
and perpendicular widths
$$\left(\frac{100}{2\sqrt2}-25,\ 10,10,10,10,10,\ \frac{100}{2\sqrt2}-25\right).$$
Numerically solving
$$\sum_i d_i \tan\theta_i=\frac{100}{\sqrt2}, \qquad \sin\theta_i = k v_i$$
gives
$$k \approx 0.0841375064631515.$$
The total travel time is then
$$T=\sum_i \frac{d_i}{v_i\cos\theta_i}$$
which evaluates to
$$T \approx 13.1265108586.$$
Rounded to 10 decimal places:
Answer: 13.1265108586