Suppose you're given six stakes and an unlimited length of string. Your objective is to plant the stakes in a flat patch of ground in such a way that you can wrap the string around the stakes in different ways to create simple polygons with integral (i.e. integer-valued) areas (see Fig. 2).
There are some restrictions:
- This is a 2D problem.
- No three stakes (or more) may be colinear. A line drawn through any two stakes must not pass through any other stakes (see Fig. 3).
- Areas must be circumscribed exactly.
- The stakes are ideal vertices (having zero cross-sectional area), and the string may only create perfectly straight, ideal edges between the stakes.
- The string must visit each stake once and only once, and terminate where it begins (thus forming a closed polygon).
- Stake coordinates should be expressed in meters (m). The stakes can be positioned at any real coordinates so long as all stake coordinates are unique.
- All six stakes must be used.
There are two separate puzzle objectives:
The accepted answer will go to the first (correct) answer that gives grid coordinates that can produce at least 6 consecutive integral areas ≤ 20 m2. (For example, areas of 3,4,5,6,7, and 8 m2, respectively). The answer should also specify the winding order of the string (around the stakes) for each polygon.
For example, an answer might look like
Stake 1: (5 m,2 m), Stake 2: (3 m,4 m), ...
Area 3 m2: 1 → 2 → 5 → 3 → 4 → 6 → 1
Area 4 m2: 1 → 3 → 4 → 6 → 2 → 5 → 1
It is acceptable if the stakes can produce additional polygons with areas not comprising the 6 consecutive integers in the solution.
A bounty of 100 rep will go to the (correct) answer that gives the grid coordinates and winding rules that can produce the greatest number of consecutive integral areas ≤ 20 m2.
In the event of a tie, the earliest solution will receive the bounty.
Good luck staking your claim. ;)
Examples of Legal and Illegal Polygons