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6

Assuming "transportation cost" means sum of distances to each of the three roads, and the side of the equilateral triangle has length $1$:


5

A solution: Proof that this is the maximum:


5

For completeness, all the solutions, excluding rotations and reflections.


3

Simlar to Dr Xorile, I think there are many solutions. Here are several:


3

This can be done with a method similar to the one I used in the prequel question. In fact, The numbers are displayed below in base 13 to keep the square-ness. (A for 10, B for 11, C for 12 and D for 13) Rotations, reflections, and permuting rows would give other solutions.


3

The solutions:


2

Maybe I've missed something, but: Then Then, Finally,


2

Observation 1 (trivial): Observation 2 (non-rigorous): Working assumption: There is a structure that precisely realizes these needs: What is missing for optimality ( or: the dangers of intuition) ? Update


2

And here are a few 2x14 solutions:


2

Here's 2 fundamentally different solutions: I suspect there are many because these are literally the first two things I tried as I was playing around and in both cases I was able to just shove the pieces in and get a solution. I did it in the numerical order shown.


1



1

Let's first work out the sizes of the rectangles. It turns out every combination of these shapes is possible.


1

This is, Method:


1

The judges can cut the cake like this: Edit: New solution for updated rules:


1

I believe I have obtained better results than those posted previously: And for the bonus question:


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