TITLE: Can the perimeter of a convex polygon X be an odd number?
QUESTION [6 upvotes]: Polygon X is a convex polygon which:

is not a triangle
has no pair of parallel sides
has all vertices with both integer coordinates
has sides with a length expressed by a positive integer

Can the perimeter of the polygon X be an odd number?
In Geogebra I have already drawn several dozen of such figures, but each of them had an even perimeter. I have no idea what I can do to get there.

REPLY [13 votes]: I suspect you can drop "convex" and the first two constraints and it is still true that the parity of the sum is even
Consider your polygon has sides length $z_1,z_2,\ldots,z_n$ where $z_i^2=x_i^2+y_i^2$ and $\sum x_i =  \sum y_i = 0$.  The $x_i$ and $y_i$ are the signed differences in co-ordinates between successive vertices, corresponding to the horizontal and vertical components of each side. Everything (the lengths, the co-ordinates and the difference in co-ordinates) is an integer.
Then the parity of $z_i$ is

equal to the parity of $z_i^2$
equal to the parity of $x_i^2+y_i^2$
equal to the parity of $x_i+y_i$

so the parity of the circumference $\sum z_i$ is equal to the parity of $\sum x_i +  \sum y_i$, which being $0$ is even.
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