Of all dodecagons laying in the cartesian plane, all of whose vertices are lattice points, and whose sides are of length $1, 2, 3, \dots,$ and $12$ in some order, which two have the largest and smallest area?
The smallest area I can find:
This has an area = $40$
The area of the angled section can be seen to be $ (6 \times 8 / 2) - (4 \times 3 / 2) - (3 \times 2) = 24 - 6 - 6 = 12$
The remainder can be counted with an area of $28$
$ 12 + 28 = 40$
I don't know this is the smallest – it is my smallest.
I found another solution without any angles, area $41$.
The largest area I can find (another edit):
This has an area = $378$
This was a lot more difficult than finding a smallest area.
The improved solution was found by looking for an enclosing rectangle or square that would maximise the area, comprised of the available dimensions. I found the possibilities
$26 \times 16$ (as used in an earlier post)
$25 \times 17$
$21 \times 21$
So I continued with that last one.
I then juggled around the 6 smallest dimensions (apart from $5$ which I wanted on a corner) to find the least area which would be lost by using rectangles as cut-outs, and I found that the smallest area which would be lost is $33$ from those rectangular cutouts.
Along with that are two mitres at the other corners, losing another $24 + 6 = 30$ area. $441 - 33 - 30 = 378$
Finally I juggled around these parts and the four remaining lengths to obtain this:
In detail, $(21 \times 21) - (6 \times 8 / 2)- (3 \times 4 / 2) - (3 \times 4) - (1 \times 7) - (2 \times 6) - (1 \times 2) = $
$ 441 - 24 - 6 - 12 - 7 - 12 - 2 = 378 $
I am fairly sure this is the largest possible - but I may be wrong.
The 5 and the 10 lengths are the only ones that can go diagonally.
They are the hypotenuse of Pythagorean triples $3:4:5$ and $6:8:10$.
Minimizing area, I present the "snake".
Should be smaller than the others found so far.
Generalization of solution:
After the parts 2,3 and 5 are used for the head, and 1 for the end, all the others can can be divided into pairs of a and a+2, and one pair of b and b+1. These pairs can all be steered to either direction, so the tail can be made to not collide with itself. This works for n-gons, where n is divisible by 4.
Daniel Mathias used this generalization for his hexadecagon answer
An alternative snake with the same area: