This week the #puzzle is: The Ant Goes Marching #geometry #minimum #coding

June the ant is on a cylinder. More specifically, she is on the edge of one of the cylinder’s two circular faces. Her dinner is on the edge of the opposite circular face, and all the way around on the other side of that face. The radius of the cylinder is 2 meters and its height is 2 meters.

Your job is to help June find the shortest path along the surface of the cylinder so that she can chow down as quickly as possible. What’s the length of this shortest path?

And for extra credit:

Now, June is on a hollowed-out cylinder, also known as a “cylindrical shell.” The shell’s outer radius is 2 meters and its inner radius is 1 meter. The shell is 2 meters tall. June is on the outer edge of one of the cylinder’s two flat faces. Her dinner is on the opposite face, and all the way around on the other end of that face.

Once again, your job is to help June find the shortest path along the surface of the shell so that she can chow down as quickly as possible. What’s the length of this shortest path?

The Ant Goes Marching

Solution, possibly incorrect:

Program Desmos

While I did mess around with various techniques, I ended up using a program to go through a lot of options.

• There’s an angle between the beginning point, the center of the top disk and the point, where June begins walking on the curved part of the cylinder. Similarly for the bottom disk. Assumption: I only have to look at cases where these 2 angles sum to 180° or less, and where the shortest possible path is taken on the curved part.
• I go through a lot of options where these 2 angles are anything from 0° to 180°.
• The length of the path across a disk with radius r (in this case 2) depends on the angle:

• For the curved part, I can simply assume, that it has been laid flat. It has height 2, and the width depends on how much is left, after the 2 angles have used some. E.g., if the 2 angles eat up π/2 between them, there’s 3π/2 left. At the most, it has width 2π.
• Add these 3 paths parts together, and we have the length of the whole path.
• Go through a lot of options, and remember the best one.

I also made a heat map of the options. It tells me, that the shortest paths occur in the corners, when 1 angle is 0 and 1 angle is 180. The program reaches the same conclusion. In this situation the path has length 6: 4 across a disk and 2 going straight down the curved part of the cylinder.

And for extra credit:

Similar to the fiddler, I use a program, with small modifications.

• I assume again there are 2 angles. This time I cap both at 90°. My program actually handles 90° slightly wrong, but the solution isn’t close to that situation, so it doesn’t matter.
• Across a disk, June travels between the outer circle and the inner. She also travels down the inner cylinder.
• For the length traveled across a disk:

• This time the solution is close to 26° for both angles. Length 5.36896.
• Heat maps. Read 540 on the left as 5.40 and 5368 on the right as 5.368. Further read 0-10 on the right as 20-30.