Daily Maths Challenges

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[solved] A tale of two rabbits (self.dailymaths)

submitted 1 week ago* (last edited 1 day ago) by joelthelion to c/dailymaths

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I have ten meters of mesh fence. I want to make two enclosures for my rabbits who can't stand each other (they can share a wall). What is the shape and the area of the largest enclosure (in terms of area) I can build? Each rabbit needs to have access to the same area. The shape can be arbitrary (although it would be nice if it were continuous or smooth to some extent, and each area contiguous).

Examples:

A square of 2x2m, divided by a 2m wall. Area: 4 square meters

A circle of radius 1.21m, divided by a wall. Area: ~4.58 square meters.

Is it possible to do better?

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[–] joelthelion 3 points 1 week ago* (last edited 1 week ago) (1 children)

It looks good, however when I compute the area, I get 4.4872..., which is less than what I get for the single circle. I could be wrong though, as others seem to be getting 4.94.

Edit: I found my mistake! I was using the wrong formula for the altitude of the equilateral triangle. Now I get 4.9458..., which is by far the best result so far. Thank you!

[–] A_A 3 points 1 week ago* (last edited 1 week ago) (2 children)

i have the same result, glad for your rabbits (and for you) !
P.S. : You can change the flare to [solved]

[–] joelthelion 2 points 1 day ago* (last edited 1 day ago)

I've implemented a gradient descent algorithm in python to optimize the shape, and I get essentially the same thing:

I'm changing the title to solved :)

Here's the code: https://pastebin.com/wKvhVvQg

[–] joelthelion 1 points 1 week ago

You have the best solution so far, but I wouldn't call it solved yet, since we don't have any proof or indication that it is the best solution that exists.