Mission: To find a minimum-length walk, for one person alone, that covers the entirety of a simple closed curve c distance units long and that begins and ends at a particular point B on that curve, given:
- An unlimited supply of rations, initially all at B, to be consumed continuously while walking.
- The ability to walk a maximum of 1 distance unit per unit of rations consumed.
- The ability to carry no more than 2 units of rations at a time.
- The ability to retrieve from any point on the curve only rations in full 1-unit packages that have remained sealed since the beginning of the mission. A sealed package must be unsealed for any of its contents to be consumed.
Michiel de Bondt seems to have solved this problem for c=5 (basically Exploring Mount Neverest by Henry Ernest Dudeney and H. F. Heath), but I am having trouble with some of de Bondt's unstated assumptions.
For example, a walk might include reaching a certain point P away from B, first from one side and later from the other side, and then returning to B. Together, B and P divide the curve into 2 parts: p1, which is completed first, and p2, which is completed second.
My question is: Is it safe to exclude from consideration any itinerary in which rations arriving at P via p1 are consumed while walking on p2?
The answer is probably obvious, but I want to know why it is obvious.
No, it is not obvious. Consider the case $c=3$ and let $P$ be one unit clockwise from $B$. I claim the following is optimal, though there are equally efficient solutions. Set out clockwise from $B$ with two rations. Drop one at $P$, eat the other, and return to $B$. Set out counterclockwise from $B$ with two rations. Eat one at one unit, get to $P$, eat the ration that you dropped there, and continue to $B$. Another solution is the mirror image. Another is after dropping the ration at $P$ to set out clockwise with two rations, get to $P$, eat a ration and pick up the dropped one, and continue around.
With the requirement to eat continuously, let $c=3$ and place $P$ at $1/2$ clockwise. You can start from $B$, drop one at $P$ and return, then start from $B$ going counterclockwise, drop one at $5/2$ and return. Now you can start from $B$ and go either direction. When you have traveled $1/2$ drop the ration you are eating and start on the one you find. Your $2$ rations will take you to the other one you dropped and you can continue to $B$.
For larger $c$ it may be useful to solve the problem on a line of length $c/2$ starting from one end. What is the most efficient way to get to the end and return to $B$? You can then make a circle by putting two of these together. It seems you could generally make the final run by going all the way around the circle as we saw for $c=3$.