MathDB

We have a regular

n{}

-gon, with

n\geqslant 4

. We consider the arrangements of

n{}

numbers on its vertices, each of which is equal to 1 or 2. For each such arrangement

K{}

, we find the number of odd sums among all sums of numbers in several consecutive vertices. This number is denoted by

\alpha(K)

.
[*]Find the largest possible value of

\alpha(K)

. [*]Find the number of arrangements for which

\alpha(K)

takes this largest possible value.
Proposed by P. Kozhevnikov

Problem Statement