Let's first eliminate a few trails that can't be part of the solution.
This red trail, completed entirely within the left-hand paths, won't work. The blue trail shown would necessarily cover the remaining paths on the left-hand side. However, the blue trail could only cover one of the four numbered paths on the right-hand side, and the remaining three could not be covered by the third and final trail.
These two red trails also suffer from a similar problem. Because each leaves a T-shaped intersection of paths uncovered, both of the remaining trails must pass through that intersection. This means that both of the remaining trails must pass through the left-hand side. On the right-hand side, the red trail would cover one of the four numbered paths, leaving three paths to be covered by two trails, which is impossible.
Each of these examples shows that a trail loop must be completely contained within the right-hand side to cover two of the four paths there. So we need a trail on the left-hand side that covers enough paths and uses one of the four numbered paths on the right-hand side.
This is the trail we are looking for. The green trail will use one of the four numbered paths to complete the loop. The second trail will pass through the remaining paths on the left-hand side and use another of the numbered paths. The third trail will be contained on the right-hand side and use the remaining two numbered paths.
There are several possible combinations that follow this idea, however. The following combinations are organized by pairs of possible green trails, going from shortest to longest. Remember, we need three trails that are at most a mile apart in length. The unique solution is outlined.
Last Updated : June 14, 2016
Knossos Games by Tim Boester
