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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