proof of the pattern

[full aspen]

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[flint dawn]

it is not so difficult to prove that for α = 4n + 2, 4n + 3, it is false
you can justify that the total "lengths" of the lines, that is, the number of segments between the points they connect, has the same parity as α
by for example, starting with all α lines connecting 2 adjacent vertices, the total "length" will be α
then you can reach any line configuration by switching 2 vertices that 2 lines connect to, which will change the total length by an even number
and you can also prove that only for α = 4n, 4n + 1 does 1 + 2 + 3 + ... + α have the same parity as α, so for 4n + 2, 4n + 3, it is definitely false

i don't know how to prove there is always a configuration for 4n, 4n + 1

for this, the longest segment has some random length involving trig/sqrts/etc based on the side length of the shape, right? so don't use that, just say its length is the number of segments (in this case 5 on each side, so 5) between the points it connects

alr

oh is parity the confusing part here
parity means evenness/oddness
if α is even, the total length is even, if odd, odd

so

you start in this configuration with total length α

then you are going to switch some lines around

like the following

start at some random configuration

swap

and this adds or removes the same amount of length from both lines

which means the total increase or decrease in length is even

[flint dawn]

here, each line is length 1

yes, and the total length = α as well

mhhh
i'm just saying if you start at the trivial lines where each point connects to the one next to it, and you make some swaps until you reach a potential working configuration, if it starts even it will remain even the whole time and if it starts odd it will remain odd the whole time

so 1 + 2 + 3 + 4 + 5 better be the same oddness as α, 5
and it in fact is
both are odd

2 diagonals on a square is not a valid configuration

so are you agreeing that my argument works

alr

yeah
you can show that 4n + 2 has diff parity from (4n+2)(4n+3)/2, etc

all 4 cases