MathDB

Let

n \geq 2

be a natural number, and let

\left( a_{1};\;a_{2};\;...;\;a_{n}\right)

be a permutation of

\left(1;\;2;\;...;\;n\right)

. For any integer

k

with

1 \leq k \leq n

, we place

a_k

raisins on the position

k

of the real number axis. [The real number axis is the

x

-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions

x_A

x_B

x_C

, each of the numbers

x_A

x_B

x_C

being an element of

\left\{1;\;2;\;...;\;n\right\}

. [It is not forbidden to place different children on the same place!] For any

k

, the

a_k

raisins placed on the position

k

are equally handed out to those children whose positions are next to

k

. [So, if there is only one child lying next to

k

, then he gets the raisin. If there are two children lying next to

k

(either both on the same position or symmetric with respect to

k

), then each of them gets one half of the raisin. Etc..] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which

n

does there exist a configuration

\left( a_{1};\;a_{2};\;...;\;a_{n}\right)

and numbers

x_A

x_B

x_C

, such that all three children are happy?

Problem Statement