Henry Ernest Dudeney/Modern Puzzles/165 - The Despatch-Rider in Flanders/Solution

by : $165$

 * The Despatch-Rider in Flanders

Solution
The rider needs to ride across the sand to a point on the turf / sand boundary exactly $1$ mile from the western boundary of the map, then straight across the turf to $B$.

Proof
Let $G$ be the point on the turf / sand boundary to which he must aim.

Let $x$ be the distance from that point to the western boundary of the map $E$.


 * Dudeney-Modern-Puzzles-165-solution.png

There appears to be no quick and easy way to solve this quartic, unless one were to plough through Ferrari's Method.

However, if we are sharp we note that:
 * $3 - 42 + 179 + 56 - 196 = 0$

and so from Coefficients of Polynomial add to 0 iff 1 is a Root we have that $x = 1$ is a root of that quartic.

We can factor that quartic if we like:
 * $\paren {x - 1} \paren {3 x^3 - 39 x^2 + 140 x + 196} = 0$

but we already have the solution $x = 1$ which adequately fulfils the conditions of the problem.

We check the geometry:

and all is well.