The origin of this question is that I've found a way to 'weight off' vertex weights from weighted $ K_n $ d. h., if one assumes that the weight $ w_ {ij} $ the edge $ e_ {ij} $ can be expressed as $ pi_i + omega_ {ij} + pi_j $ with vertex potentials $ pi_i $ and $ pi_j $is it then possible to restore that $ omega_ {ij} $,

I was then curious to rediscover the vertex potentials and to this end set up a system of linear equations obtained from the edges of a Hamilton cycle.

After a corresponding renaming of the vertices, the equations have the form

$ pi_i + pi_ {i + 1} = w_ {i, i + 1} – omega_ {i, i + 1}; quad 0 le i le , quad n + 1: = 0 $

To my great surprise, it first turned out that one has one for the determinant of the associated matrix

$$ begin {vmatrix}

1 & 1 & 0 & 0 & 0 & 0 & Points & 0 & 0 \

0 & 1 & 1 & 0 & 0 & cdots & 0 & 0 \

0 & 0 & 1 & 1 & 0 & cdots & 0 & 0 \

vdots & vdots & ddots & ddots & ddots & ddots & vdots & vdots \

0 & 0 & Points & 0 & 1 & 1 & 0 & 0 \

0 & 0 & cdots & 0 & 0 & 1 & 1 & 0 \

1 & 0 & points & 0 & 0 & 0 & 0 & 1

end {vmatrix} = 0 iff n equiv0 mod2 $$ This means that the unknown vertex potentials can only be determined if their number is odd.

After some reflection on this observation, it became clear that this is the determinant of analog matrices $ m $ consecutive $ 1 $Instead of two, cyclically shifted from right to right from one row to the next by two, s does not appear to be exactly zero when $ m $ and $ n $ are relatively prime.

Question:Is there a non-trivial explanation for the phenomenon of the solvability of related problems, the only noticeable difference being the number $ n $ of equations depends on the number of theoretical properties of the relationship between $ n $ and a fixed set of parameters that are independent of $ n $?

The arguments that I imagine would be based only on the "original" message of the problem and not on its "translation" into a system of linear equations.

Ideally, the arguments with no background in linear algebra would be understandable.