A physical geoboard is an organized set pegs that are distributed in a grid pattern which sits on (or is a part of) a thin rectangular base. Different sized Rubber bands can wrap around the pegs of geoboard, which can form a large number of patterns and shapes. An example of this is shown in the picture below.
When I was a kid I had a computer program that was a digital version of a geoboard. The program had a square grid of nodes. The grid was ten nodes by ten nodes. Nodes on the grid could be connected by line segments.
One feature of the digital geoboard is that if a closed shape was formed by line segments, the perimeter of the shape could be measured. One unit on the geoboard was the distance between two adjacent points in the horizontal or vertical direction.
Here is a YouTube video of the program.
One thing I tried to do during the time when I had the program was to find the largest perimeter closed shape that I could create with the grid. After a lot of trial and error I eventually came up this:
If the dot coordinates are defined by $(x,y)$ and the bottom left node is defined as $(1,1)$ then the polygon in the picture directly above can be defined by the connected nodes $(1,10)\rightarrow (1,9)\rightarrow (2,9)\rightarrow (1,8)\rightarrow (3,9)\rightarrow (1,7)\rightarrow (4,9)\rightarrow (1,6)\rightarrow (5,9)\rightarrow (1,5)\rightarrow (6,9)\rightarrow (1,4)\rightarrow (7,9)\rightarrow (1,3)\rightarrow (8,9)\rightarrow (1,2)\rightarrow (9,9)\rightarrow (1,1)\rightarrow (9,8)\rightarrow (2,1)\rightarrow (9,7)\rightarrow (3,1)\rightarrow (9,6)\rightarrow (4,1)\rightarrow (9,5)\rightarrow (5,1)\rightarrow (9,4)\rightarrow (6,1)\rightarrow (9,3)\rightarrow (7,1)\rightarrow (9,2)\rightarrow (8,1)\rightarrow (10,1)\rightarrow (10,10)\rightarrow (1,10)$
This polygon has a polygon has a perimeter of $\approx 202.32\space units$
The exact amount is presented by the following expression $$20+8\sqrt{2}+2\sum_{n=1}^{15}\sqrt{\lceil\frac{n^2}{2}}\rceil$$
At the time I came up with this I thought that this wasn't the absolute maximum, but It was very close to the maximum and I was quite satisfied with result.
This problem has never completely left my mind and much later I realized that I could do a lot better. The polygon directly below is my best thus far.
This is defined by the connected points $(1,10)\rightarrow (1,9)\rightarrow (2,9)\rightarrow (1,8)\rightarrow (3,9)\rightarrow (2,8)\rightarrow (4,9)\rightarrow (1,7)\rightarrow (3,8)\rightarrow (2,7)\rightarrow (5,9)\rightarrow (1,6)\rightarrow (4,8)\rightarrow (3,7)\rightarrow (6,9)\rightarrow (2,6)\rightarrow (5,8)\rightarrow (1,5)\rightarrow (4,7)\rightarrow (3,6)\rightarrow (7,9)\rightarrow (2,5)\rightarrow (6,8)\rightarrow (1,4)\rightarrow (5,7)\rightarrow (4,6)\rightarrow (8,9)\rightarrow (3,5)\rightarrow (7,8)\rightarrow (2,4)\rightarrow (6,7)\rightarrow (1,3)\rightarrow (5,6)\rightarrow (4,5)\rightarrow (9,9)\rightarrow (3,4)\rightarrow (8,8)\rightarrow (2,3)\rightarrow (7,7)\rightarrow (1,2)\rightarrow (6,6)\rightarrow (5,5)\rightarrow (9,8)\rightarrow (4,4)\rightarrow (8,7)\rightarrow (3,3)\rightarrow (7,6)\rightarrow (2,2)\rightarrow (6,5)\rightarrow (1,1)\rightarrow (5,4)\rightarrow (4,3)\rightarrow (9,7)\rightarrow (3,2)\rightarrow (8,6)\rightarrow (2,1)\rightarrow (7,5)\rightarrow (6,4)\rightarrow (9,6)\rightarrow (5,3)\rightarrow (8,5)\rightarrow (4,2)\rightarrow (7,4)\rightarrow (3,1)\rightarrow (6,3)\rightarrow (5,2)\rightarrow (9,5)\rightarrow (4,1)\rightarrow (8,4)\rightarrow (7,3)\rightarrow (9,4)\rightarrow (6,2)\rightarrow (8,3)\rightarrow (5,1)\rightarrow (7,2)\rightarrow (6,1)\rightarrow (9,3)\rightarrow (7,1)\rightarrow (9,2)\rightarrow (8,1)\rightarrow (10,1)\rightarrow (10,10)\rightarrow (1,10)$
This polygon has a perimeter of $\approx 355.06 \space units$
The exact amount is presented by the following expression $$122+16\sqrt{2}+7\sqrt{5}+13\sqrt{13}+17\sqrt{41}+5\sqrt{61}$$
Question: Can we find and prove a maximum for the ten by ten grid? If so can we generalize this strategy to find the maximum for the $N$ x $N$ grid?
Just for clarification on the rules, the only two requirements is that the line segment end points must be on the nodes whose coordinates are positive integers (within the square) and the shape must be closed. A shape like the one shown in the picture below is allowed, however only the distances between the intersection points are counted toward the perimeter of the shape.
from Hot Weekly Questions - Mathematics Stack Exchange
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