# Can the perimeter of a convex polygon X be an odd number?

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.

• You need to use Pythagorean triplets to construct a polygon. You can start at the origin $(0, 0)$. Somehow you have to be able to close the polygon, keep it convex, and make sure no parallel sides--I don't have much advice on that. Commented Dec 6, 2021 at 13:49

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.

• [+1] Excellent. Commented Dec 6, 2021 at 14:07
• @SaucyO'Path I took the $(x_i, y_i)$ to be vector components of the sides (so they add up to 0 because it's a closed polygon). Commented Dec 6, 2021 at 14:17
• @SaucyO'Path 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 Commented Dec 6, 2021 at 14:19
• @Henry Ah, that's right!
– user562983
Commented Dec 6, 2021 at 14:21